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Номер 6.2, страница 33 - гдз по геометрии 7 класс учебник Смирнов, Туяков
Авторы: Смирнов В. А., Туяков Е. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2017 - 2026
ISBN: 978-601-07-0873-0
Утверждено Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 7 классе
Глава 1. Начальные геометрические сведения. Параграф 6. Измерение величин углов - номер 6.2, страница 33.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129502 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/06-2" "field_display_title" => "6.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129359 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Начальные геометрические сведения" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1047 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1078 #items: array:1 [ 0 => App\Models\Book {#1049 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 2907 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1051 #items: array:1 [ 0 => App\Models\Term {#1048 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "геометрию" "field_creative_case" => "геометрией" "field_dative_case" => "геометрии" "field_genitive_case" => "геометрии" "field_nominative_case" => "геометрия" "field_prepositional_case" => "геометрии" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #original: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "геометрию" "field_creative_case" => "геометрией" "field_dative_case" => "геометрии" "field_genitive_case" => "геометрии" "field_nominative_case" => "геометрия" "field_prepositional_case" => "геометрии" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ 0 => App\Models\Term {#1053 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5456 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "7" "field_cases" => array:6 [ "field_accusative_case" => "седьмой" "field_creative_case" => "седьмым" "field_dative_case" => "седьмому" "field_genitive_case" => "седьмого" "field_nominative_case" => "седьмой" "field_prepositional_case" => "седьмом" ] "field_translit" => "sedmoj" ] #original: array:6 [ "id" => 5456 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "7" "field_cases" => array:6 [ "field_accusative_case" => "седьмой" "field_creative_case" => "седьмым" "field_dative_case" => "седьмому" "field_genitive_case" => "седьмого" "field_nominative_case" => "седьмой" "field_prepositional_case" => "седьмом" ] "field_translit" => "sedmoj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ 0 => App\Models\Term {#1055 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:2 [ 0 => App\Models\Term {#1057 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 5801 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Смирнов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Владимир" "field_patronymic" => "Алексеевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 5801 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Смирнов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Владимир" "field_patronymic" => "Алексеевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1058 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7017 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Туяков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Есенкельды" "field_patronymic" => "Алыбаевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 7017 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Туяков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Есенкельды" "field_patronymic" => "Алыбаевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060 #items: array:1 [ 0 => App\Models\Term {#1061 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1062 #items: array:1 [ 0 => App\Models\Term {#1063 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2017" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 7 класса общеобразовательной школы" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "528" "field_priority" => "2" "field_default_folder" => "/geometrija_07/Smirnova-u17/" "field_isbn" => "978-601-07-0873-0" "field_cover" => array:1 [ 0 => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 378 "subject" => 543 "class_subject" => 389 ] ] #original: array:50 [ "id" => 2907 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1051} "field_class" => Illuminate\Database\Eloquent\Collection {#1052} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054} "field_author" => Illuminate\Database\Eloquent\Collection {#1056} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1059} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060} "field_country" => Illuminate\Database\Eloquent\Collection {#1062} "field_city" => Illuminate\Database\Eloquent\Collection {#1064} "field_series" => Illuminate\Database\Eloquent\Collection {#1066} "field_umk" => Illuminate\Database\Eloquent\Collection {#1067} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1068} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1071} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2017" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 7 класса общеобразовательной школы" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "528" "field_priority" => "2" "field_default_folder" => "/geometrija_07/Smirnova-u17/" "field_isbn" => "978-601-07-0873-0" "field_cover" => array:1 [ 0 => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_07/Smirnova-u17/covers/cover1.webp?ts=1753259592" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1072} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1073} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1074} "field_url" => "/7-klass/geometrija/smirnova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1075} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 378 "subject" => 543 "class_subject" => 389 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1129359 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Начальные геометрические сведения" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1078} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1105 #items: array:2 [ 0 => App\Models\Branch {#1113 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129359 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Начальные геометрические сведения" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1114 #items: array:1 [ 0 => App\Models\Term {#1047} ] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1115 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1116 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1129359 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Начальные геометрические сведения" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1114} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1115} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1116} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1117 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129365 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Измерение величин углов" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ 0 => App\Models\Term {#1118 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "31" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1120 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1129365 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Измерение величин углов" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1121} "field_page_start" => "31" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1120} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1119} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:3 [ 0 => App\Models\Element {#1095 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1319572 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ 0 => App\Models\Edition {#1094 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1092 #items: array:1 [ 0 => App\Models\Term {#1093 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1090 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1092} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1090} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1129502" "type" => "task" ] "text" => "<p><strong>6.2.</strong> Найдите величины углов $AOB$, $AOC$, $AOD$, $BOC$, $BOD$, $COD$, изображенных на рисунке 6.3.</p><p>1:1000</p><p>Рис. 6.3</p>" "img" => array:2 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1771" "height" => 207 "path" => "/media/geometrija_07/Smirnova-u17/0-06/2-1.webp?ts=1753982245" ] 1 => array:5 [ "name" => "2-2.jpg" "alt" => null "width" => "1113" "height" => 764 "path" => "/media/geometrija_07/Smirnova-u17/0-06/2-2.webp?ts=1753982245" ] ] ] #original: array:7 [ "id" => 1319572 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087} "task" => array:2 [ "refs" => "1129502" "type" => "task" ] "text" => "<p><strong>6.2.</strong> Найдите величины углов $AOB$, $AOC$, $AOD$, $BOC$, $BOD$, $COD$, изображенных на рисунке 6.3.</p><p>1:1000</p><p>Рис. 6.3</p>" "img" => array:2 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1771" "height" => 207 "path" => "/media/geometrija_07/Smirnova-u17/0-06/2-1.webp?ts=1753982245" ] 1 => array:5 [ "name" => "2-2.jpg" "alt" => null "width" => "1113" "height" => 764 "path" => "/media/geometrija_07/Smirnova-u17/0-06/2-2.webp?ts=1753982245" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1089 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1320025 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Edition {#1086 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5010 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Term {#1085 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6714 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gdz" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "gdz" ] #original: array:6 [ "id" => 6714 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gdz" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null …4 ] "field_translit" => "gdz" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5010 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1082} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1084} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1083} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1080} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1129502" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1476" "height" => 686 "path" => "/media/geometrija_07/Smirnova-u17/1-06/2-1.webp?ts=1753982363" ] ] ] #original: array:6 [ "id" => 1320025 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1079} "task" => array:2 [ "refs" => "1129502" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1476" "height" => 686 "path" => "/media/geometrija_07/Smirnova-u17/1-06/2-1.webp?ts=1753982363" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1081 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1541362 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Edition {#1077 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5631 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1041 #items: array:1 [ 0 => App\Models\Term {#1043 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7012 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gemini 2.5 Pro" "field_cases" => array:6 [ …6] "field_translit" => "gemini 2.5 pro" ] #original: array:6 [ "id" => 7012 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gemini 2.5 Pro" "field_cases" => array:6 [ …6] "field_translit" => "gemini 2.5 pro" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1034 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5631 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1041} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1034} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1039} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1129502" "type" => "task" ] "text" => "<p>Для определения величин углов воспользуемся транспортиром, изображенным на рисунке. Центр транспортира находится в точке O. Будем использовать внутреннюю шкалу, где луч OA соответствует отметке 0°.</p><svg width="450" height="250" viewBox="0 0 450 250" xmlns="http://www.w3.org/2000/svg"><defs><style> .label { font-family: sans-serif; font-size: 16px; } .scale-text { font-family: sans-serif; font-size: 12px; text-anchor: middle; } </style></defs><path d="M 25 225 L 425 225 L 425 220 A 200 200 0 0 0 25 220 Z" fill="#FBEBDD" stroke="#6D5545" stroke-width="1.5"/><g stroke="#6D5545" stroke-width="1"><path d="M 425 220 L 425 205" /><path d="M 25 220 L 25 205" /><path d="M 225 20 L 225 35" /><path d="M 378.2 91.4 L 370.5 97.8" /><path d="M 293.4 32 L 286.9 38.4" /><path d="M 37.2 151.6 L 44.8 157.9" /></g><text x="425" y="195" class="scale-text">0</text><text x="395" y="115" class="scale-text">40</text><text x="310" y="55" class="scale-text">70</text><text x="225" y="15" class="scale-text">90</text><text x="60" y="145" class="scale-text">160</text><text x="25" y="195" class="scale-text">180</text><line x1="225" y1="220" x2="435" y2="220" stroke="red" stroke-width="2"/><line x1="225" y1="220" x2="388.2" y2="81.4" stroke="red" stroke-width="2"/><line x1="225" y1="220" x2="298.4" y2="22" stroke="red" stroke-width="2"/><line x1="225" y1="220" x2="27.2" y2="141.6" stroke="red" stroke-width="2"/><circle cx="225" cy="220" r="4" fill="red"/><text x="210" y="215" class="label">O</text><text x="440" y="225" class="label">A</text><text x="395" y="75" class="label">B</text><text x="300" y="15" class="label">C</text><text x="15" y="140" class="label">D</text></svg><p>Сначала определим положение каждого луча на шкале транспортира:</p><p>• Луч OA проходит через отметку $0^\circ$.</p><p>• Луч OB проходит через отметку $40^\circ$.</p><p>• Луч OC проходит через отметку $70^\circ$.</p><p>• Луч OD проходит через отметку $160^\circ$.</p><p><strong>AOB</strong></p><p>Величина угла $\angle AOB$ равна разности показаний лучей OB и OA на шкале транспортира.</p><p>$\angle AOB = 40^\circ - 0^\circ = 40^\circ$</p><p>Ответ: $40^\circ$.</p><p><strong>AOC</strong></p><p>Величина угла $\angle AOC$ равна разности показаний лучей OC и OA на шкале транспортира.</p><p>$\angle AOC = 70^\circ - 0^\circ = 70^\circ$</p><p>Ответ: $70^\circ$.</p><p><strong>AOD</strong></p><p>Величина угла $\angle AOD$ равна разности показаний лучей OD и OA на шкале транспортира.</p><p>$\angle AOD = 160^\circ - 0^\circ = 160^\circ$</p><p>Ответ: $160^\circ$.</p><p><strong>BOC</strong></p><p>Величина угла $\angle BOC$ равна разности показаний лучей OC и OB. Также можно найти этот угол как разность углов $\angle AOC$ и $\angle AOB$.</p><p>$\angle BOC = 70^\circ - 40^\circ = 30^\circ$</p><p>Ответ: $30^\circ$.</p><p><strong>BOD</strong></p><p>Величина угла $\angle BOD$ равна разности показаний лучей OD и OB. Также можно найти этот угол как разность углов $\angle AOD$ и $\angle AOB$.</p><p>$\angle BOD = 160^\circ - 40^\circ = 120^\circ$</p><p>Ответ: $120^\circ$.</p><p><strong>COD</strong></p><p>Величина угла $\angle COD$ равна разности показаний лучей OD и OC. Также можно найти этот угол как разность углов $\angle AOD$ и $\angle AOC$.</p><p>$\angle COD = 160^\circ - 70^\circ = 90^\circ$</p><p>Ответ: $90^\circ$.</p>" ] #original: array:6 [ "id" => 1541362 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1129502" "type" => "task" ] "text" => "<p>Для определения величин углов воспользуемся транспортиром, изображенным на рисунке. Центр транспортира находится в точке O. Будем использовать внутреннюю шкалу, где луч OA соответствует отметке 0°.</p><svg width="450" height="250" viewBox="0 0 450 250" xmlns="http://www.w3.org/2000/svg"><defs><style> .label { font-family: sans-serif; font-size: 16px; } .scale-text { font-family: sans-serif; font-size: 12px; text-anchor: middle; } </style></defs><path d="M 25 225 L 425 225 L 425 220 A 200 200 0 0 0 25 220 Z" fill="#FBEBDD" stroke="#6D5545" stroke-width="1.5"/><g stroke="#6D5545" stroke-width="1"><path d="M 425 220 L 425 205" /><path d="M 25 220 L 25 205" /><path d="M 225 20 L 225 35" /><path d="M 378.2 91.4 L 370.5 97.8" /><path d="M 293.4 32 L 286.9 38.4" /><path d="M 37.2 151.6 L 44.8 157.9" /></g><text x="425" y="195" class="scale-text">0</text><text x="395" y="115" class="scale-text">40</text><text x="310" y="55" class="scale-text">70</text><text x="225" y="15" class="scale-text">90</text><text x="60" y="145" class="scale-text">160</text><text x="25" y="195" class="scale-text">180</text><line x1="225" y1="220" x2="435" y2="220" stroke="red" stroke-width="2"/><line x1="225" y1="220" x2="388.2" y2="81.4" stroke="red" stroke-width="2"/><line x1="225" y1="220" x2="298.4" y2="22" stroke="red" stroke-width="2"/><line x1="225" y1="220" x2="27.2" y2="141.6" stroke="red" stroke-width="2"/><circle cx="225" cy="220" r="4" fill="red"/><text x="210" y="215" class="label">O</text><text x="440" y="225" class="label">A</text><text x="395" y="75" class="label">B</text><text x="300" y="15" class="label">C</text><text x="15" y="140" class="label">D</text></svg><p>Сначала определим положение каждого луча на шкале транспортира:</p><p>• Луч OA проходит через отметку $0^\circ$.</p><p>• Луч OB проходит через отметку $40^\circ$.</p><p>• Луч OC проходит через отметку $70^\circ$.</p><p>• Луч OD проходит через отметку $160^\circ$.</p><p><strong>AOB</strong></p><p>Величина угла $\angle AOB$ равна разности показаний лучей OB и OA на шкале транспортира.</p><p>$\angle AOB = 40^\circ - 0^\circ = 40^\circ$</p><p>Ответ: $40^\circ$.</p><p><strong>AOC</strong></p><p>Величина угла $\angle AOC$ равна разности показаний лучей OC и OA на шкале транспортира.</p><p>$\angle AOC = 70^\circ - 0^\circ = 70^\circ$</p><p>Ответ: $70^\circ$.</p><p><strong>AOD</strong></p><p>Величина угла $\angle AOD$ равна разности показаний лучей OD и OA на шкале транспортира.</p><p>$\angle AOD = 160^\circ - 0^\circ = 160^\circ$</p><p>Ответ: $160^\circ$.</p><p><strong>BOC</strong></p><p>Величина угла $\angle BOC$ равна разности показаний лучей OC и OB. Также можно найти этот угол как разность углов $\angle AOC$ и $\angle AOB$.</p><p>$\angle BOC = 70^\circ - 40^\circ = 30^\circ$</p><p>Ответ: $30^\circ$.</p><p><strong>BOD</strong></p><p>Величина угла $\angle BOD$ равна разности показаний лучей OD и OB. Также можно найти этот угол как разность углов $\angle AOD$ и $\angle AOB$.</p><p>$\angle BOD = 160^\circ - 40^\circ = 120^\circ$</p><p>Ответ: $120^\circ$.</p><p><strong>COD</strong></p><p>Величина угла $\angle COD$ равна разности показаний лучей OD и OC. Также можно найти этот угол как разность углов $\angle AOD$ и $\angle AOC$.</p><p>$\angle COD = 160^\circ - 70^\circ = 90^\circ$</p><p>Ответ: $90^\circ$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129503" "type" => "task" ] "previous" => array:2 [ "refs" => "1129501" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1098 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\BookPage {#1042 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1114350 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_url" => "/7-klass/geometrija/smirnova-uchebnik/page-33" "field_display_title" => "33" "field_folder" => "1" "field_image_name" => "33" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1040 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1103 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1112 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\Element {#1127 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1290537 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1129 #items: array:1 [ 0 => App\Models\Edition {#1094} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1114350" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "33-1.jpg" "alt" => null "width" => "1905" "height" => 2809 "path" => "/media/geometrija_07/Smirnova-u17/0-1/33-1.webp?ts=1753260140" ] ] ] #original: array:6 [ "id" => 1290537 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1129} "book_page" => array:2 [ "refs" => "1114350" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "33-1.jpg" "alt" => null "width" => "1905" "height" => 2809 "path" => "/media/geometrija_07/Smirnova-u17/0-1/33-1.webp?ts=1753260140" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1114351" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114349" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1197 #items: array:3 [ 0 => App\Models\Task {#1207 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129502 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/06-2" "field_display_title" => "6.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1208 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1209 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1210 #items: array:1 [ 0 => App\Models\Branch {#1113} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:2 [ 0 => App\Models\Branch {#1113} 1 => App\Models\Branch {#1117} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1212 #items: array:3 [ 0 => App\Models\Element {#1095} 1 => App\Models\Element {#1089} 2 => App\Models\Element {#1081} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129503" "type" => "task" ] "previous" => array:2 [ "refs" => "1129501" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1213 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114350" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129502 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/06-2" "field_display_title" => "6.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1208} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1209} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1210} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1211} "content" => Illuminate\Database\Eloquent\Collection {#1212} "next" => array:2 [ "refs" => "1129503" "type" => "task" ] "previous" => array:2 [ "refs" => "1129501" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1213} "page" => array:2 [ "refs" => "1114350" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1214 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129503 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/06-3" "field_display_title" => "6.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1215 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1216 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:1 [ 0 => App\Models\Branch {#1113} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1218 #items: array:2 [ 0 => App\Models\Branch {#1113} 1 => App\Models\Branch {#1117} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1221 #items: array:3 [ 0 => App\Models\Element {#1230 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1319573 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1231 …2} "task" => array:2 [ …2] "text" => "<p><strong>6.3.</strong> С помощью транспортира постройте углы величиной $10^\circ$, $30^\circ$, $70^\circ$, $100^\circ$, $150^\circ$.</p><p><em>Рис. 6.3</em></p>" "img" => array:1 [ …1] ] #original: array:7 [ "id" => 1319573 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1231 …2} "task" => array:2 [ …2] "text" => "<p><strong>6.3.</strong> С помощью транспортира постройте углы величиной $10^\circ$, $30^\circ$, $70^\circ$, $100^\circ$, $150^\circ$.</p><p><em>Рис. 6.3</em></p>" "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1232 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1320027 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1233 …2} "task" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1320027 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1233 …2} "task" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1234 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1541363 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1235 …2} "task" => array:2 [ …2] "text" => "<p>Для построения угла заданной величины с помощью транспортира необходимо выполнить следующие шаги. Ниже представлен общий алгоритм, который затем применяется для построения каждого из требуемых углов.</p><p><strong>Общий алгоритм построения угла:</strong></p><ol><li>Начертите на плоскости произвольный луч, например, луч $OA$. Точка $O$ будет вершиной будущего угла, а луч $OA$ — одной из его сторон.</li><li>Приложите транспортир так, чтобы его центр совпал с точкой $O$ (вершиной угла), а прямая линия основания транспортира (отметка $0^\circ$) прошла точно по лучу $OA$.</li><li>Найдите на шкале транспортира деление, соответствующее заданной величине угла. Важно использовать ту шкалу (внутреннюю или внешнюю), которая начинается с $0^\circ$ на вашем луче $OA$.</li><li>Поставьте точку (назовем ее $B$) карандашом напротив этого деления на бумаге.</li><li>Уберите транспортир и с помощью линейки соедините вершину $O$ с отмеченной точкой $B$, проведя второй луч $OB$.</li><li>Полученный угол $AOB$ и будет искомым углом.</li></ol><p>Теперь построим каждый из заданных углов, следуя этому алгоритму.</p><p><strong>Построение угла $10^\circ$</strong></p><p>Для построения угла в $10^\circ$, мы следуем общему алгоритму. Сначала чертим луч $OA$. Затем прикладываем транспортир центром к точке $O$ и совмещаем его основание с лучом $OA$. На шкале, которая начинается с нуля на луче $OA$, находим отметку $10$. Ставим напротив нее точку $B$. Убрав транспортир, проводим луч $OB$. Угол $AOB$ и есть искомый угол в $10^\circ$.</p><svg width="250" height="180" viewBox="0 0 250 180" xmlns="http://www.w3.org/2000/svg"><g stroke="black" stroke-width="1.5" font-family="sans-serif" font-size="16"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">A</text><text x="105" y="155">O</text><g transform="rotate(-10, 120, 150)"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">B</text></g></g><path d="M 150 150 A 30 30 0 0 1 149.55 144.78" stroke="red" stroke-width="1.5" fill="none"></path><text x="155" y="140" fill="red" font-size="14">10°</text></svg><p>Ответ: Угол величиной $10^\circ$ построен в соответствии с описанными шагами.</p><p><strong>Построение угла $30^\circ$</strong></p><p>Аналогично предыдущему пункту, строим луч $OA$. Прикладываем транспортир, совмещая центр с точкой $O$ и нулевую линию с лучом $OA$. Находим на шкале отметку $30$, ставим точку $B$ и проводим луч $OB$. Полученный угол $AOB$ равен $30^\circ$.</p><svg width="250" height="180" viewBox="0 0 250 180" xmlns="http://www.w3.org/2000/svg"><g stroke="black" stroke-width="1.5" font-family="sans-serif" font-size="16"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">A</text><text x="105" y="155">O</text><g transform="rotate(-30, 120, 150)"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">B</text></g></g><path d="M 150 150 A 30 30 0 0 1 145.98 135" stroke="red" stroke-width="1.5" fill="none"></path><text x="155" y="130" fill="red" font-size="14">30°</text></svg><p>Ответ: Острый угол величиной $30^\circ$ построен.</p><p><strong>Построение угла $70^\circ$</strong></p><p>Выполняем шаги алгоритма для величины $70^\circ$. Чертим луч $OA$. Прикладываем транспортир. Находим на шкале отметку $70$, ставим точку $B$ и проводим луч $OB$. Угол $AOB$ равен $70^\circ$.</p><svg width="250" height="180" viewBox="0 0 250 180" xmlns="http://www.w3.org/2000/svg"><g stroke="black" stroke-width="1.5" font-family="sans-serif" font-size="16"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">A</text><text x="105" y="155">O</text><g transform="rotate(-70, 120, 150)"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">B</text></g></g><path d="M 150 150 A 30 30 0 0 1 130.26 121.8" stroke="red" stroke-width="1.5" fill="none"></path><text x="145" y="115" fill="red" font-size="14">70°</text></svg><p>Ответ: Острый угол величиной $70^\circ$ построен.</p><p><strong>Построение угла $100^\circ$</strong></p><p>Действуем аналогично. Последовательно выполняем шаги: чертим луч $OA$; прикладываем транспортир к точке $O$ и лучу $OA$; находим на шкале отметку $100$; ставим точку $B$; проводим луч $OB$. Так как $100^\circ > 90^\circ$, полученный угол $AOB$ является тупым.</p><svg width="250" height="180" viewBox="0 0 250 180" xmlns="http://www.w3.org/2000/svg"><g stroke="black" stroke-width="1.5" font-family="sans-serif" font-size="16"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">A</text><text x="105" y="155">O</text><g transform="rotate(-100, 120, 150)"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">B</text></g></g><path d="M 150 150 A 30 30 0 0 1 114.78 120.45" stroke="red" stroke-width="1.5" fill="none"></path><text x="125" y="120" fill="red" font-size="14">100°</text></svg><p>Ответ: Тупой угол величиной $100^\circ$ построен.</p><p><strong>Построение угла $150^\circ$</strong></p><p>Для построения угла $150^\circ$ повторяем те же действия. Чертим луч $OA$, прикладываем транспортир, находим на шкале отметку $150$, отмечаем точку $B$ и соединяем ее с вершиной $O$. Полученный тупой угол $AOB$ будет равен $150^\circ$.</p><svg width="250" height="180" viewBox="0 0 250 180" xmlns="http://www.w3.org/2000/svg"><g stroke="black" stroke-width="1.5" font-family="sans-serif" font-size="16"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">A</text><text x="105" y="155">O</text><g transform="rotate(-150, 120, 150)"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">B</text></g></g><path d="M 150 150 A 30 30 0 1 1 94.02 135" stroke="red" stroke-width="1.5" fill="none"></path><text x="90" y="125" fill="red" font-size="14">150°</text></svg><p>Ответ: Тупой угол величиной $150^\circ$ построен.</p>" ] #original: array:6 [ "id" => 1541363 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1235 …2} "task" => array:2 [ …2] "text" => "<p>Для построения угла заданной величины с помощью транспортира необходимо выполнить следующие шаги. Ниже представлен общий алгоритм, который затем применяется для построения каждого из требуемых углов.</p><p><strong>Общий алгоритм построения угла:</strong></p><ol><li>Начертите на плоскости произвольный луч, например, луч $OA$. Точка $O$ будет вершиной будущего угла, а луч $OA$ — одной из его сторон.</li><li>Приложите транспортир так, чтобы его центр совпал с точкой $O$ (вершиной угла), а прямая линия основания транспортира (отметка $0^\circ$) прошла точно по лучу $OA$.</li><li>Найдите на шкале транспортира деление, соответствующее заданной величине угла. Важно использовать ту шкалу (внутреннюю или внешнюю), которая начинается с $0^\circ$ на вашем луче $OA$.</li><li>Поставьте точку (назовем ее $B$) карандашом напротив этого деления на бумаге.</li><li>Уберите транспортир и с помощью линейки соедините вершину $O$ с отмеченной точкой $B$, проведя второй луч $OB$.</li><li>Полученный угол $AOB$ и будет искомым углом.</li></ol><p>Теперь построим каждый из заданных углов, следуя этому алгоритму.</p><p><strong>Построение угла $10^\circ$</strong></p><p>Для построения угла в $10^\circ$, мы следуем общему алгоритму. Сначала чертим луч $OA$. Затем прикладываем транспортир центром к точке $O$ и совмещаем его основание с лучом $OA$. На шкале, которая начинается с нуля на луче $OA$, находим отметку $10$. Ставим напротив нее точку $B$. Убрав транспортир, проводим луч $OB$. Угол $AOB$ и есть искомый угол в $10^\circ$.</p><svg width="250" height="180" viewBox="0 0 250 180" xmlns="http://www.w3.org/2000/svg"><g stroke="black" stroke-width="1.5" font-family="sans-serif" font-size="16"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">A</text><text x="105" y="155">O</text><g transform="rotate(-10, 120, 150)"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">B</text></g></g><path d="M 150 150 A 30 30 0 0 1 149.55 144.78" stroke="red" stroke-width="1.5" fill="none"></path><text x="155" y="140" fill="red" font-size="14">10°</text></svg><p>Ответ: Угол величиной $10^\circ$ построен в соответствии с описанными шагами.</p><p><strong>Построение угла $30^\circ$</strong></p><p>Аналогично предыдущему пункту, строим луч $OA$. Прикладываем транспортир, совмещая центр с точкой $O$ и нулевую линию с лучом $OA$. Находим на шкале отметку $30$, ставим точку $B$ и проводим луч $OB$. Полученный угол $AOB$ равен $30^\circ$.</p><svg width="250" height="180" viewBox="0 0 250 180" xmlns="http://www.w3.org/2000/svg"><g stroke="black" stroke-width="1.5" font-family="sans-serif" font-size="16"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">A</text><text x="105" y="155">O</text><g transform="rotate(-30, 120, 150)"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">B</text></g></g><path d="M 150 150 A 30 30 0 0 1 145.98 135" stroke="red" stroke-width="1.5" fill="none"></path><text x="155" y="130" fill="red" font-size="14">30°</text></svg><p>Ответ: Острый угол величиной $30^\circ$ построен.</p><p><strong>Построение угла $70^\circ$</strong></p><p>Выполняем шаги алгоритма для величины $70^\circ$. Чертим луч $OA$. Прикладываем транспортир. Находим на шкале отметку $70$, ставим точку $B$ и проводим луч $OB$. Угол $AOB$ равен $70^\circ$.</p><svg width="250" height="180" viewBox="0 0 250 180" xmlns="http://www.w3.org/2000/svg"><g stroke="black" stroke-width="1.5" font-family="sans-serif" font-size="16"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">A</text><text x="105" y="155">O</text><g transform="rotate(-70, 120, 150)"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">B</text></g></g><path d="M 150 150 A 30 30 0 0 1 130.26 121.8" stroke="red" stroke-width="1.5" fill="none"></path><text x="145" y="115" fill="red" font-size="14">70°</text></svg><p>Ответ: Острый угол величиной $70^\circ$ построен.</p><p><strong>Построение угла $100^\circ$</strong></p><p>Действуем аналогично. Последовательно выполняем шаги: чертим луч $OA$; прикладываем транспортир к точке $O$ и лучу $OA$; находим на шкале отметку $100$; ставим точку $B$; проводим луч $OB$. Так как $100^\circ > 90^\circ$, полученный угол $AOB$ является тупым.</p><svg width="250" height="180" viewBox="0 0 250 180" xmlns="http://www.w3.org/2000/svg"><g stroke="black" stroke-width="1.5" font-family="sans-serif" font-size="16"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">A</text><text x="105" y="155">O</text><g transform="rotate(-100, 120, 150)"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">B</text></g></g><path d="M 150 150 A 30 30 0 0 1 114.78 120.45" stroke="red" stroke-width="1.5" fill="none"></path><text x="125" y="120" fill="red" font-size="14">100°</text></svg><p>Ответ: Тупой угол величиной $100^\circ$ построен.</p><p><strong>Построение угла $150^\circ$</strong></p><p>Для построения угла $150^\circ$ повторяем те же действия. Чертим луч $OA$, прикладываем транспортир, находим на шкале отметку $150$, отмечаем точку $B$ и соединяем ее с вершиной $O$. Полученный тупой угол $AOB$ будет равен $150^\circ$.</p><svg width="250" height="180" viewBox="0 0 250 180" xmlns="http://www.w3.org/2000/svg"><g stroke="black" stroke-width="1.5" font-family="sans-serif" font-size="16"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">A</text><text x="105" y="155">O</text><g transform="rotate(-150, 120, 150)"><line x1="120" y1="150" x2="220" y2="150"></line><text x="225" y="155">B</text></g></g><path d="M 150 150 A 30 30 0 1 1 94.02 135" stroke="red" stroke-width="1.5" fill="none"></path><text x="90" y="125" fill="red" font-size="14">150°</text></svg><p>Ответ: Тупой угол величиной $150^\circ$ построен.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129504" "type" => "task" ] "previous" => array:2 [ "refs" => "1129502" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1227 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114350" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129503 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/06-3" "field_display_title" => "6.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1215} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1216} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1217} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1218} "content" => Illuminate\Database\Eloquent\Collection {#1221} "next" => array:2 [ "refs" => "1129504" "type" => "task" ] "previous" => array:2 [ "refs" => "1129502" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1227} "page" => array:2 [ "refs" => "1114350" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1220 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1129504 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/06-4" "field_display_title" => "6.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1226 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1228 #items: array:1 [ 0 => App\Models\Branch {#1113} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1224 #items: array:2 [ 0 => App\Models\Branch {#1113} 1 => App\Models\Branch {#1117} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:3 [ 0 => App\Models\Element {#1244 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1319574 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1245 …2} "task" => array:2 [ …2] "text" => "<p><strong>6.4.</strong> На клетчатой бумаге изобразите луч $AB$, как показано на рисунке 6.4. От луча $AB$ отложите угол $BAC$, равный:</p><p><strong>а) $45^\circ$</strong></p><p><strong>б) $90^\circ$</strong></p><p>а)</p><p>б)</p><p>Рис. 6.4</p>" "img" => array:1 [ …1] ] #original: array:7 [ "id" => 1319574 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1245 …2} "task" => array:2 [ …2] "text" => "<p><strong>6.4.</strong> На клетчатой бумаге изобразите луч $AB$, как показано на рисунке 6.4. От луча $AB$ отложите угол $BAC$, равный:</p><p><strong>а) $45^\circ$</strong></p><p><strong>б) $90^\circ$</strong></p><p>а)</p><p>б)</p><p>Рис. 6.4</p>" "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1246 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1320029 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1247 …2} "task" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1320029 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1247 …2} "task" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1248 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1541364 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1249 …2} "task" => array:2 [ …2] "text" => "<p><strong>а)</strong> На рисунке а) луч $AB$ проходит через узлы сетки таким образом, что для перехода из точки $A$ в точку $B$ необходимо сместиться на 3 клетки вправо и на 3 клетки вверх. Такой луч образует угол $45^{\circ}$ с горизонтальной линией сетки, проходящей через точку $A$.</p><p>Чтобы отложить от луча $AB$ угол $\angle BAC$, равный $45^{\circ}$, необходимо построить луч $AC$, который будет образовывать с лучом $AB$ угол в $45^{\circ}$. Существует два возможных положения для луча $AC$:</p><ol><li>Луч $AC$ можно провести так, чтобы угол между ним и горизонталью составлял $45^{\circ} + 45^{\circ} = 90^{\circ}$. В этом случае луч $AC$ будет направлен вертикально вверх.</li><li>Луч $AC$ можно провести так, чтобы угол между ним и горизонталью составлял $45^{\circ} - 45^{\circ} = 0^{\circ}$. В этом случае луч $AC$ будет направлен горизонтально вправо.</li></ol><p>Построим второй вариант, проведя луч $AC$ вдоль горизонтальной линии сетки вправо от точки $A$.</p><svg width="180" height="150" viewBox="0 0 180 150" xmlns="http://www.w3.org/2000/svg"><defs><style>.grid { stroke: #add8e6; stroke-width: 1; } .ray { stroke: red; stroke-width: 2; } .point { fill: red; } .label { font-family: 'Times New Roman', serif; font-size: 16px; font-style: italic; }</style><marker id="arrow_a" viewBox="0 0 10 10" refX="8" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse"><path d="M 0 0 L 10 5 L 0 10 z" fill="red"></path></marker></defs><!-- Grid --><path d="M 10 10 H 160 M 10 40 H 160 M 10 70 H 160 M 10 100 H 160 M 10 130 H 160" class="grid"></path><path d="M 10 10 V 130 M 40 10 V 130 M 70 10 V 130 M 100 10 V 130 M 130 10 V 130 M 160 10 V 130" class="grid"></path><!-- Original Ray AB --><line x1="40" y1="100" x2="130" y2="10" class="ray"></line><circle cx="40" cy="100" r="3.5" class="point"></circle><text x="30" y="118" class="label">A</text><text x="135" y="15" class="label">B</text><!-- Constructed Ray AC --><line x1="40" y1="100" x2="160" y2="100" class="ray" marker-end="url(#arrow_a)"></line><text x="165" y="105" class="label">C</text><!-- Angle Arc --><path d="M 70 100 A 30 30 0 0 0 61.21 78.79" fill="none" stroke="black" stroke-width="1"></path></svg><p>Ответ: На рисунке показано одно из возможных построений угла $\angle BAC = 45^{\circ}$.</p><p><strong>б)</strong> На рисунке б) луч $AB$ проходит через узлы сетки так, что для перехода из точки $A$ в точку $B$ необходимо сместиться на 4 клетки вправо и на 2 клетки вверх. Чтобы построить угол $\angle BAC$, равный $90^{\circ}$, нужно провести луч $AC$, перпендикулярный лучу $AB$.</p><p>Для построения перпендикулярного луча на клетчатой бумаге можно использовать правило векторов. Если вектор, соответствующий лучу $AB$, имеет координаты $(\Delta x, \Delta y)$, то перпендикулярный ему вектор будет иметь координаты $(-\Delta y, \Delta x)$ или $(\Delta y, -\Delta x)$.</p><p>Для луча $AB$ смещение от $A$ составляет $\Delta x = 4$ клетки и $\Delta y = 2$ клетки. Следовательно, для перпендикулярного луча $AC$ смещение от точки $A$ может быть:</p><ol><li>$(-\Delta y, \Delta x) = (-2, 4)$, то есть 2 клетки влево и 4 клетки вверх.</li><li>$(\Delta y, -\Delta x) = (2, -4)$, то есть 2 клетки вправо и 4 клетки вниз.</li></ol><p>Оба направления задают лучи, перпендикулярные $AB$. Выберем второе направление и проведем луч $AC$. Любая точка на этом луче, например, полученная смещением на 1 клетку вправо и 2 клетки вниз от точки $A$, будет задавать нужный луч.</p><svg width="180" height="150" viewBox="0 0 180 150" xmlns="http://www.w3.org/2000/svg"><defs><style>.grid { stroke: #add8e6; stroke-width: 1; } .ray { stroke: red; stroke-width: 2; } .point { fill: red; } .label { font-family: 'Times New Roman', serif; font-size: 16px; font-style: italic; }</style><marker id="arrow_b" viewBox="0 0 10 10" refX="8" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse"><path d="M 0 0 L 10 5 L 0 10 z" fill="red"></path></marker></defs><!-- Grid --><path d="M 10 10 H 160 M 10 40 H 160 M 10 70 H 160 M 10 100 H 160 M 10 130 H 160" class="grid"></path><path d="M 10 10 V 130 M 40 10 V 130 M 70 10 V 130 M 100 10 V 130 M 130 10 V 130 M 160 10 V 130" class="grid"></path><!-- Original Ray AB --><line x1="40" y1="70" x2="160" y2="10" class="ray"></line><circle cx="40" cy="70" r="3.5" class="point"></circle><text x="30" y="88" class="label">A</text><text x="160" y="15" class="label">B</text><!-- Constructed Ray AC --><line x1="40" y1="70" x2="70" y2="130" class="ray" marker-end="url(#arrow_b)"></line><text x="75" y="135" class="label">C</text><!-- Angle Arc --><path d="M 57.89 61.06 L 66.83 78.95 L 48.94 87.89" fill="none" stroke="black" stroke-width="1"></path></svg><p>Ответ: На рисунке показано одно из возможных построений угла $\angle BAC = 90^{\circ}$.</p>" ] #original: array:6 [ "id" => 1541364 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1249 …2} "task" => array:2 [ …2] "text" => "<p><strong>а)</strong> На рисунке а) луч $AB$ проходит через узлы сетки таким образом, что для перехода из точки $A$ в точку $B$ необходимо сместиться на 3 клетки вправо и на 3 клетки вверх. Такой луч образует угол $45^{\circ}$ с горизонтальной линией сетки, проходящей через точку $A$.</p><p>Чтобы отложить от луча $AB$ угол $\angle BAC$, равный $45^{\circ}$, необходимо построить луч $AC$, который будет образовывать с лучом $AB$ угол в $45^{\circ}$. Существует два возможных положения для луча $AC$:</p><ol><li>Луч $AC$ можно провести так, чтобы угол между ним и горизонталью составлял $45^{\circ} + 45^{\circ} = 90^{\circ}$. В этом случае луч $AC$ будет направлен вертикально вверх.</li><li>Луч $AC$ можно провести так, чтобы угол между ним и горизонталью составлял $45^{\circ} - 45^{\circ} = 0^{\circ}$. В этом случае луч $AC$ будет направлен горизонтально вправо.</li></ol><p>Построим второй вариант, проведя луч $AC$ вдоль горизонтальной линии сетки вправо от точки $A$.</p><svg width="180" height="150" viewBox="0 0 180 150" xmlns="http://www.w3.org/2000/svg"><defs><style>.grid { stroke: #add8e6; stroke-width: 1; } .ray { stroke: red; stroke-width: 2; } .point { fill: red; } .label { font-family: 'Times New Roman', serif; font-size: 16px; font-style: italic; }</style><marker id="arrow_a" viewBox="0 0 10 10" refX="8" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse"><path d="M 0 0 L 10 5 L 0 10 z" fill="red"></path></marker></defs><!-- Grid --><path d="M 10 10 H 160 M 10 40 H 160 M 10 70 H 160 M 10 100 H 160 M 10 130 H 160" class="grid"></path><path d="M 10 10 V 130 M 40 10 V 130 M 70 10 V 130 M 100 10 V 130 M 130 10 V 130 M 160 10 V 130" class="grid"></path><!-- Original Ray AB --><line x1="40" y1="100" x2="130" y2="10" class="ray"></line><circle cx="40" cy="100" r="3.5" class="point"></circle><text x="30" y="118" class="label">A</text><text x="135" y="15" class="label">B</text><!-- Constructed Ray AC --><line x1="40" y1="100" x2="160" y2="100" class="ray" marker-end="url(#arrow_a)"></line><text x="165" y="105" class="label">C</text><!-- Angle Arc --><path d="M 70 100 A 30 30 0 0 0 61.21 78.79" fill="none" stroke="black" stroke-width="1"></path></svg><p>Ответ: На рисунке показано одно из возможных построений угла $\angle BAC = 45^{\circ}$.</p><p><strong>б)</strong> На рисунке б) луч $AB$ проходит через узлы сетки так, что для перехода из точки $A$ в точку $B$ необходимо сместиться на 4 клетки вправо и на 2 клетки вверх. Чтобы построить угол $\angle BAC$, равный $90^{\circ}$, нужно провести луч $AC$, перпендикулярный лучу $AB$.</p><p>Для построения перпендикулярного луча на клетчатой бумаге можно использовать правило векторов. Если вектор, соответствующий лучу $AB$, имеет координаты $(\Delta x, \Delta y)$, то перпендикулярный ему вектор будет иметь координаты $(-\Delta y, \Delta x)$ или $(\Delta y, -\Delta x)$.</p><p>Для луча $AB$ смещение от $A$ составляет $\Delta x = 4$ клетки и $\Delta y = 2$ клетки. Следовательно, для перпендикулярного луча $AC$ смещение от точки $A$ может быть:</p><ol><li>$(-\Delta y, \Delta x) = (-2, 4)$, то есть 2 клетки влево и 4 клетки вверх.</li><li>$(\Delta y, -\Delta x) = (2, -4)$, то есть 2 клетки вправо и 4 клетки вниз.</li></ol><p>Оба направления задают лучи, перпендикулярные $AB$. Выберем второе направление и проведем луч $AC$. Любая точка на этом луче, например, полученная смещением на 1 клетку вправо и 2 клетки вниз от точки $A$, будет задавать нужный луч.</p><svg width="180" height="150" viewBox="0 0 180 150" xmlns="http://www.w3.org/2000/svg"><defs><style>.grid { stroke: #add8e6; stroke-width: 1; } .ray { stroke: red; stroke-width: 2; } .point { fill: red; } .label { font-family: 'Times New Roman', serif; font-size: 16px; font-style: italic; }</style><marker id="arrow_b" viewBox="0 0 10 10" refX="8" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse"><path d="M 0 0 L 10 5 L 0 10 z" fill="red"></path></marker></defs><!-- Grid --><path d="M 10 10 H 160 M 10 40 H 160 M 10 70 H 160 M 10 100 H 160 M 10 130 H 160" class="grid"></path><path d="M 10 10 V 130 M 40 10 V 130 M 70 10 V 130 M 100 10 V 130 M 130 10 V 130 M 160 10 V 130" class="grid"></path><!-- Original Ray AB --><line x1="40" y1="70" x2="160" y2="10" class="ray"></line><circle cx="40" cy="70" r="3.5" class="point"></circle><text x="30" y="88" class="label">A</text><text x="160" y="15" class="label">B</text><!-- Constructed Ray AC --><line x1="40" y1="70" x2="70" y2="130" class="ray" marker-end="url(#arrow_b)"></line><text x="75" y="135" class="label">C</text><!-- Angle Arc --><path d="M 57.89 61.06 L 66.83 78.95 L 48.94 87.89" fill="none" stroke="black" stroke-width="1"></path></svg><p>Ответ: На рисунке показано одно из возможных построений угла $\angle BAC = 90^{\circ}$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1129505" "type" => "task" ] "previous" => array:2 [ "refs" => "1129503" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1241 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1114350" "type" => "book_page" ] ] #original: array:24 [ "id" => 1129504 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/7-klass/geometrija/smirnova-uchebnik/06-4" "field_display_title" => "6.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1219} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1226} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1228} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1224} "content" => Illuminate\Database\Eloquent\Collection {#1223} "next" => array:2 [ "refs" => "1129505" "type" => "task" ] "previous" => array:2 [ "refs" => 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[] #guarded: array:1 [ 0 => "*" ] }
№6.2 (с. 33)
Условие. №6.2 (с. 33)
скриншот условия
6.2. Найдите величины углов $AOB$, $AOC$, $AOD$, $BOC$, $BOD$, $COD$, изображенных на рисунке 6.3.
1:1000
Рис. 6.3
Решение. №6.2 (с. 33)
Решение 2. №6.2 (с. 33)
Для определения величин углов воспользуемся транспортиром, изображенным на рисунке. Центр транспортира находится в точке O. Будем использовать внутреннюю шкалу, где луч OA соответствует отметке 0°.
Сначала определим положение каждого луча на шкале транспортира:
• Луч OA проходит через отметку $0^\circ$.
• Луч OB проходит через отметку $40^\circ$.
• Луч OC проходит через отметку $70^\circ$.
• Луч OD проходит через отметку $160^\circ$.
AOB
Величина угла $\angle AOB$ равна разности показаний лучей OB и OA на шкале транспортира.
$\angle AOB = 40^\circ - 0^\circ = 40^\circ$
Ответ: $40^\circ$.
AOC
Величина угла $\angle AOC$ равна разности показаний лучей OC и OA на шкале транспортира.
$\angle AOC = 70^\circ - 0^\circ = 70^\circ$
Ответ: $70^\circ$.
AOD
Величина угла $\angle AOD$ равна разности показаний лучей OD и OA на шкале транспортира.
$\angle AOD = 160^\circ - 0^\circ = 160^\circ$
Ответ: $160^\circ$.
BOC
Величина угла $\angle BOC$ равна разности показаний лучей OC и OB. Также можно найти этот угол как разность углов $\angle AOC$ и $\angle AOB$.
$\angle BOC = 70^\circ - 40^\circ = 30^\circ$
Ответ: $30^\circ$.
BOD
Величина угла $\angle BOD$ равна разности показаний лучей OD и OB. Также можно найти этот угол как разность углов $\angle AOD$ и $\angle AOB$.
$\angle BOD = 160^\circ - 40^\circ = 120^\circ$
Ответ: $120^\circ$.
COD
Величина угла $\angle COD$ равна разности показаний лучей OD и OC. Также можно найти этот угол как разность углов $\angle AOD$ и $\angle AOC$.
$\angle COD = 160^\circ - 70^\circ = 90^\circ$
Ответ: $90^\circ$.
Другие задания:
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