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Номер 278, страница 85 - гдз по математике 5 класс учебник Бунимович, Дорофеев
Авторы: Бунимович Е. А., Дорофеев Г. В., Суворова С. Б., Кузнецова Л. В., Минаева С. С., Рослова Л. О.
Тип: Учебник
Серия: сферы
Издательство: Просвещение
Год издания: 2023 - 2026
Уровень обучения: базовый
Цвет обложки: оранжевый с компасом
ISBN: 978-5-09-105798-0
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Глава 5. Углы и многоугольники. 17. Как обозначают и сравнивают углы - номер 278, страница 85.
App\Models\Task {#1024 // 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" => 534474 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_page_end" => null "field_url" => "/5-klass/matematika/bunimovich-uchebnik/00-278" "field_display_title" => "278" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #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 {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #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" => 534087 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Углы и многоугольники" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: array:1 [ 0 => App\Models\Term {#1033 #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 [ …6] "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 [ …6] "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" => "81" "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 {#1030 #items: array:1 [ 0 => App\Models\Book {#1032 #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" => 59 "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 {#1034 #items: array:1 [ 0 => App\Models\Term {#1035 #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" => 6484 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Математика" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "matematika" ] #original: array:10 [ "id" => 6484 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Математика" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "matematika" ] #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 {#1036 #items: array:1 [ 0 => App\Models\Term {#1037 #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" => 5445 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "5" "field_cases" => array:6 [ …6] "field_translit" => "pjatyj" ] #original: array:6 [ "id" => 5445 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "5" "field_cases" => array:6 [ …6] "field_translit" => "pjatyj" ] #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 {#1038 #items: array:1 [ 0 => App\Models\Term {#1039 #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" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #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 {#1040 #items: array:6 [ 0 => App\Models\Term {#1041 #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" => 3941 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бунимович" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Евгений" "field_patronymic" => "Абрамович" "field_surname_rp" => null ] #original: array:12 [ "id" => 3941 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бунимович" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Евгений" "field_patronymic" => "Абрамович" "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 {#1042 #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" => 4283 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Дорофеев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Георгий" "field_patronymic" => "Владимирович" "field_surname_rp" => null ] #original: array:12 [ "id" => 4283 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Дорофеев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Георгий" "field_patronymic" => "Владимирович" "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 => "*" ] } 2 => 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:12 [ "id" => 5879 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Суворова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Светлана" "field_patronymic" => "Борисовна" "field_surname_rp" => null ] #original: array:12 [ "id" => 5879 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Суворова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Светлана" "field_patronymic" => "Борисовна" "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 => "*" ] } 3 => App\Models\Term {#1044 #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" => 4848 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кузнецова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Людмила" "field_patronymic" => "Викторовна" "field_surname_rp" => null ] #original: array:12 [ "id" => 4848 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кузнецова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Людмила" "field_patronymic" => "Викторовна" "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 => "*" ] } 4 => App\Models\Term {#1045 #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" => 5198 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Минаева" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Светлана" "field_patronymic" => "Станиславовна" "field_surname_rp" => null ] #original: array:12 [ "id" => 5198 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Минаева" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Светлана" "field_patronymic" => "Станиславовна" "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 => "*" ] } 5 => App\Models\Term {#1046 #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" => 5622 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Рослова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Лариса" "field_patronymic" => "Олеговна" "field_surname_rp" => null ] #original: array:12 [ "id" => 5622 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Рослова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Лариса" "field_patronymic" => "Олеговна" "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 {#1047 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:1 [ 0 => App\Models\Term {#1049 #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 [ …6] "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 [ …6] "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 {#1050 #items: array:1 [ 0 => App\Models\Term {#1051 #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" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #original: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #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 {#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" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #original: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #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 {#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:8 [ "id" => 4795 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "сферы" "field_cases" => null "field_foreign_lang_series" => null "field_transcription_eng" => null "field_translit" => "sfery" ] #original: array:8 [ "id" => 4795 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "сферы" "field_cases" => null "field_foreign_lang_series" => null "field_transcription_eng" => null "field_translit" => "sfery" ] #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_umk" => Illuminate\Database\Eloquent\Collection {#1056 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:1 [ 0 => 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:6 [ "id" => 38 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "базовый" "field_cases" => array:6 [ …6] "field_translit" => "bazovyy" ] #original: array:6 [ "id" => 38 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "базовый" "field_cases" => array:6 [ …6] "field_translit" => "bazovyy" ] #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_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1059 #items: array:1 [ 0 => App\Models\Term {#1060 #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" => 25 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (новый)" "field_cases" => array:6 [ …6] "field_color" => "красный" "field_end_year" => "2027" "field_start_year" => "2023" "field_translit" => "fgos" "field_type" => "новый" ] #original: array:10 [ "id" => 25 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (новый)" "field_cases" => array:6 [ …6] "field_color" => "красный" "field_end_year" => "2027" "field_start_year" => "2023" "field_translit" => "fgos" "field_type" => "новый" ] #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_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "с компасом" "field_publication_year" => "2023" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "941" "field_priority" => "10" "field_default_folder" => "/matematika_05/bunimovich-u23/" "field_isbn" => "978-5-09-105798-0" "field_cover" => array:1 [ 0 => "/media/matematika_05/bunimovich-u23/covers/cover1.webp?ts=1737321399" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/matematika_05/bunimovich-u23/covers/cover1.webp?ts=1737321399" "alt" => "" "width" => "1200" "height" => "1604" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_url" => "/5-klass/matematika/bunimovich-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1066 #items: array:1 [ 0 => App\Models\Term {#1067 #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" => 50 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "оранжевый" "field_cases" => array:6 [ …6] "field_translit" => "oranzhevyy" ] #original: array:6 [ "id" => 50 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "оранжевый" "field_cases" => array:6 [ …6] "field_translit" => "oranzhevyy" ] #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 "breadcrumbs" => array:3 [ "class" => 376 "subject" => 544 "class_subject" => 398 ] ] #original: array:50 [ "id" => 59 "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 {#1034} "field_class" => Illuminate\Database\Eloquent\Collection {#1036} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1038} "field_author" => Illuminate\Database\Eloquent\Collection {#1040} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1047} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1048} "field_country" => Illuminate\Database\Eloquent\Collection {#1050} "field_city" => Illuminate\Database\Eloquent\Collection {#1052} "field_series" => Illuminate\Database\Eloquent\Collection {#1054} "field_umk" => Illuminate\Database\Eloquent\Collection {#1056} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1057} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1059} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1061} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1062} "field_under_the_edition_degree" => null "field_cover_description" => "с компасом" "field_publication_year" => "2023" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "941" "field_priority" => "10" "field_default_folder" => "/matematika_05/bunimovich-u23/" "field_isbn" => "978-5-09-105798-0" "field_cover" => array:1 [ 0 => "/media/matematika_05/bunimovich-u23/covers/cover1.webp?ts=1737321399" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/matematika_05/bunimovich-u23/covers/cover1.webp?ts=1737321399" "alt" => "" "width" => "1200" "height" => "1604" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1063} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1064} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1065} "field_url" => "/5-klass/matematika/bunimovich-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1066} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 376 "subject" => 544 "class_subject" => 398 ] ] #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 {#1068 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 534087 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Углы и многоугольники" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "field_page_start" => "81" "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 {#1030} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068} ] #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 {#1069 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1070 #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" => 534088 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "17. Как обозначают и сравнивают углы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "82" "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 {#1072 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Branch {#1074 #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" => 534087 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Углы и многоугольники" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1075 …2} "field_page_start" => "81" "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 {#1077 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1113 …2} ] #original: array:24 [ "id" => 534087 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Углы и многоугольники" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1075 …2} "field_page_start" => "81" "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 {#1077 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1113 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 534088 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "17. Как обозначают и сравнивают углы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1071} "field_page_start" => "82" "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 {#1072} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1073} ] #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 {#1114 #items: array:2 [ 0 => App\Models\Element {#1115 #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" => 646346 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1117 #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" => 1542 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1118 …2} "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" => "margarita" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_content_source" => null ] #original: array:21 [ "id" => 1542 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1118 …2} "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" => "margarita" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …2} "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" => "534474" "type" => "task" ] "text" => "<p><strong>278 1)</strong> Начертите угол $BOC$. Постройте угол $AOB$, дополняющий его до развёрнутого угла. Постройте угол $DOC$, дополняющий угол $BOC$ до развёрнутого.</p><p><strong>2)</strong> Каким является угол $AOB$, если угол $BOC$ острый; прямой; тупой?</p><p><strong>3)</strong> Верно ли, что углы $AOB$ и $DOC$ равны? Почему?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "278-1.jpg" "alt" => null "width" => "2035" "height" => 370 "path" => "/media/matematika_05/bunimovich-u23/0-00/278-1.webp?ts=1737374198" ] ] ] #original: array:7 [ "id" => 646346 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "534474" "type" => "task" ] "text" => "<p><strong>278 1)</strong> Начертите угол $BOC$. Постройте угол $AOB$, дополняющий его до развёрнутого угла. Постройте угол $DOC$, дополняющий угол $BOC$ до развёрнутого.</p><p><strong>2)</strong> Каким является угол $AOB$, если угол $BOC$ острый; прямой; тупой?</p><p><strong>3)</strong> Верно ли, что углы $AOB$ и $DOC$ равны? Почему?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "278-1.jpg" "alt" => null "width" => "2035" "height" => 370 "path" => "/media/matematika_05/bunimovich-u23/0-00/278-1.webp?ts=1737374198" ] ] ] #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 {#1123 #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" => 1468758 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1124 #items: array:1 [ 0 => App\Models\Edition {#1125 #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" => 5408 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1126 …2} "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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5408 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1126 …2} "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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1130 …2} "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" => "534474" "type" => "task" ] "text" => "<p><strong>1)</strong></p><p>Чтобы выполнить построение, следуйте шагам:</p><ol><li>Начертите произвольный угол $BOC$. Он имеет вершину в точке $O$ и стороны, образованные лучами $OB$ и $OC$.</li> <li>Чтобы построить угол $AOB$, дополняющий угол $BOC$ до развёрнутого, необходимо из вершины $O$ провести луч $OA$ так, чтобы он являлся продолжением луча $OC$. В результате лучи $OA$ и $OC$ образуют прямую линию $AC$. Углы $AOB$ и $BOC$ будут смежными, а их сумма будет равна $180^\circ$.</li> <li>Аналогично, чтобы построить угол $DOC$, дополняющий угол $BOC$ до развёрнутого, необходимо из вершины $O$ провести луч $OD$ так, чтобы он являлся продолжением луча $OB$. В результате лучи $OD$ и $OB$ образуют прямую линию $DB$. Углы $DOC$ и $BOC$ также будут смежными, и их сумма будет равна $180^\circ$.</li></ol><p>В результате такого построения получатся две прямые, $AC$ и $BD$, пересекающиеся в точке $O$.</p><p>Ответ: Построение описано выше. В результате получаются две пересекающиеся в точке $O$ прямые $AC$ и $BD$.</p><p><strong>2)</strong></p><p>Углы $AOB$ и $BOC$ дополняют друг друга до развёрнутого угла, значит, они являются смежными. Сумма смежных углов равна $180^\circ$.</p><p>$\angle AOB + \angle BOC = 180^\circ$</p><p>Из этого следует, что $\angle AOB = 180^\circ - \angle BOC$.</p><p>Рассмотрим три случая в зависимости от вида угла $BOC$:</p><ul><li>Если угол $BOC$ — <b>острый</b> ($0^\circ < \angle BOC < 90^\circ$), то $\angle AOB = 180^\circ - \angle BOC$ будет больше $90^\circ$ и меньше $180^\circ$. Следовательно, угол $AOB$ — <b>тупой</b>.</li> <li>Если угол $BOC$ — <b>прямой</b> ($\angle BOC = 90^\circ$), то $\angle AOB = 180^\circ - 90^\circ = 90^\circ$. Следовательно, угол $AOB$ — также <b>прямой</b>.</li> <li>Если угол $BOC$ — <b>тупой</b> ($90^\circ < \angle BOC < 180^\circ$), то $\angle AOB = 180^\circ - \angle BOC$ будет больше $0^\circ$ и меньше $90^\circ$. Следовательно, угол $AOB$ — <b>острый</b>.</li></ul><p>Ответ: Если угол $BOC$ острый, то угол $AOB$ — тупой; если угол $BOC$ прямой, то угол $AOB$ — прямой; если угол $BOC$ тупой, то угол $AOB$ — острый.</p><p><strong>3)</strong></p><p>Да, утверждение, что углы $AOB$ и $DOC$ равны, верно. Вот почему:</p><p>По условию, угол $AOB$ дополняет угол $BOC$ до развёрнутого, что математически записывается как:</p><p>$\angle AOB + \angle BOC = 180^\circ$</p><p>Также по условию, угол $DOC$ дополняет угол $BOC$ до развёрнутого, что записывается как:</p><p>$\angle DOC + \angle BOC = 180^\circ$</p><p>Мы имеем два выражения, которые равны $180^\circ$. Следовательно, мы можем приравнять их левые части:</p><p>$\angle AOB + \angle BOC = \angle DOC + \angle BOC$</p><p>Вычтем из обеих частей равенства величину угла $BOC$:</p><p>$\angle AOB = \angle DOC$</p><p>Это доказывает, что углы $AOB$ и $DOC$ равны. Такие углы также называют вертикальными.</p><p>Ответ: Да, верно. Углы $AOB$ и $DOC$ равны, так как каждый из них является дополнением одного и того же угла $BOC$ до $180^\circ$.</p>" ] #original: array:6 [ "id" => 1468758 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1124} "task" => array:2 [ "refs" => "534474" "type" => "task" ] "text" => "<p><strong>1)</strong></p><p>Чтобы выполнить построение, следуйте шагам:</p><ol><li>Начертите произвольный угол $BOC$. Он имеет вершину в точке $O$ и стороны, образованные лучами $OB$ и $OC$.</li> <li>Чтобы построить угол $AOB$, дополняющий угол $BOC$ до развёрнутого, необходимо из вершины $O$ провести луч $OA$ так, чтобы он являлся продолжением луча $OC$. В результате лучи $OA$ и $OC$ образуют прямую линию $AC$. Углы $AOB$ и $BOC$ будут смежными, а их сумма будет равна $180^\circ$.</li> <li>Аналогично, чтобы построить угол $DOC$, дополняющий угол $BOC$ до развёрнутого, необходимо из вершины $O$ провести луч $OD$ так, чтобы он являлся продолжением луча $OB$. В результате лучи $OD$ и $OB$ образуют прямую линию $DB$. Углы $DOC$ и $BOC$ также будут смежными, и их сумма будет равна $180^\circ$.</li></ol><p>В результате такого построения получатся две прямые, $AC$ и $BD$, пересекающиеся в точке $O$.</p><p>Ответ: Построение описано выше. В результате получаются две пересекающиеся в точке $O$ прямые $AC$ и $BD$.</p><p><strong>2)</strong></p><p>Углы $AOB$ и $BOC$ дополняют друг друга до развёрнутого угла, значит, они являются смежными. Сумма смежных углов равна $180^\circ$.</p><p>$\angle AOB + \angle BOC = 180^\circ$</p><p>Из этого следует, что $\angle AOB = 180^\circ - \angle BOC$.</p><p>Рассмотрим три случая в зависимости от вида угла $BOC$:</p><ul><li>Если угол $BOC$ — <b>острый</b> ($0^\circ < \angle BOC < 90^\circ$), то $\angle AOB = 180^\circ - \angle BOC$ будет больше $90^\circ$ и меньше $180^\circ$. Следовательно, угол $AOB$ — <b>тупой</b>.</li> <li>Если угол $BOC$ — <b>прямой</b> ($\angle BOC = 90^\circ$), то $\angle AOB = 180^\circ - 90^\circ = 90^\circ$. Следовательно, угол $AOB$ — также <b>прямой</b>.</li> <li>Если угол $BOC$ — <b>тупой</b> ($90^\circ < \angle BOC < 180^\circ$), то $\angle AOB = 180^\circ - \angle BOC$ будет больше $0^\circ$ и меньше $90^\circ$. Следовательно, угол $AOB$ — <b>острый</b>.</li></ul><p>Ответ: Если угол $BOC$ острый, то угол $AOB$ — тупой; если угол $BOC$ прямой, то угол $AOB$ — прямой; если угол $BOC$ тупой, то угол $AOB$ — острый.</p><p><strong>3)</strong></p><p>Да, утверждение, что углы $AOB$ и $DOC$ равны, верно. Вот почему:</p><p>По условию, угол $AOB$ дополняет угол $BOC$ до развёрнутого, что математически записывается как:</p><p>$\angle AOB + \angle BOC = 180^\circ$</p><p>Также по условию, угол $DOC$ дополняет угол $BOC$ до развёрнутого, что записывается как:</p><p>$\angle DOC + \angle BOC = 180^\circ$</p><p>Мы имеем два выражения, которые равны $180^\circ$. Следовательно, мы можем приравнять их левые части:</p><p>$\angle AOB + \angle BOC = \angle DOC + \angle BOC$</p><p>Вычтем из обеих частей равенства величину угла $BOC$:</p><p>$\angle AOB = \angle DOC$</p><p>Это доказывает, что углы $AOB$ и $DOC$ равны. Такие углы также называют вертикальными.</p><p>Ответ: Да, верно. Углы $AOB$ и $DOC$ равны, так как каждый из них является дополнением одного и того же угла $BOC$ до $180^\circ$.</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" => "534475" "type" => "task" ] "previous" => array:2 [ "refs" => "534473" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1131 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\BookPage {#1145 #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" => 949379 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_url" => "/5-klass/matematika/bunimovich-uchebnik/page-85" "field_display_title" => "85" "field_folder" => "1" "field_image_name" => "85" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1147 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1185 #items: array:1 [ 0 => App\Models\Book {#1148 #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" => 59 "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 {#1150 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1151 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1155 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1162 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1169 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1177 …2} "field_under_the_edition_degree" => null "field_cover_description" => "с компасом" "field_publication_year" => "2023" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "941" "field_priority" => "10" "field_default_folder" => "/matematika_05/bunimovich-u23/" "field_isbn" => "978-5-09-105798-0" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_url" => "/5-klass/matematika/bunimovich-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 59 "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 {#1150 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1151 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1155 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1162 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_series" => 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"field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_url" => "/5-klass/matematika/bunimovich-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 "edition_groups" => Illuminate\Database\Eloquent\Collection {#1183 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1194 #items: array:1 [ 0 => App\Models\Element {#1193 #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" => 1124661 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1203 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1124661 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1203 …2} "book_page" => 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 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "949380" "type" => "book_page" ] "previous" => array:2 [ "refs" => "949378" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1844 #items: array:7 [ 0 => App\Models\Task {#1858 #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" => 534469 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_page_end" => null "field_url" => "/5-klass/matematika/bunimovich-uchebnik/00-273" "field_display_title" => "273" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1862 …2} "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 {#1861 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1904 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2022 …2} "content" => Illuminate\Database\Eloquent\Collection {#2039 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2041 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 534469 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_page_end" => null "field_url" => "/5-klass/matematika/bunimovich-uchebnik/00-273" "field_display_title" => "273" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1862 …2} "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 {#1861 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1904 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2022 …2} "content" => Illuminate\Database\Eloquent\Collection {#2039 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2041 …2} "page" => array:2 [ …2] ] #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 {#2040 #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" => 534470 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_page_end" => null "field_url" => "/5-klass/matematika/bunimovich-uchebnik/00-274" "field_display_title" => "274" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2042 …2} "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 {#2043 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2044 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2045 …2} "content" => Illuminate\Database\Eloquent\Collection {#2062 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2064 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 534470 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_page_end" => null "field_url" => "/5-klass/matematika/bunimovich-uchebnik/00-274" "field_display_title" => "274" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2042 …2} "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 {#2043 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2044 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2045 …2} "content" => Illuminate\Database\Eloquent\Collection {#2062 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2064 …2} "page" => array:2 [ …2] ] #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 {#2063 #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" => 534471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_page_end" => null "field_url" => "/5-klass/matematika/bunimovich-uchebnik/00-275" "field_display_title" => "275" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2065 …2} "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 {#2066 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2067 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2068 …2} "content" => Illuminate\Database\Eloquent\Collection {#2085 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2087 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 534471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_page_end" => null "field_url" => "/5-klass/matematika/bunimovich-uchebnik/00-275" "field_display_title" => "275" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2065 …2} "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 {#2066 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2067 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2068 …2} "content" => Illuminate\Database\Eloquent\Collection {#2085 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2087 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#2086 #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" => 534472 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_page_end" => null "field_url" => "/5-klass/matematika/bunimovich-uchebnik/00-276" "field_display_title" => "276" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2088 …2} "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 {#2089 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2090 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2091 …2} "content" => Illuminate\Database\Eloquent\Collection {#2108 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2110 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 534472 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_page_end" => null "field_url" => "/5-klass/matematika/bunimovich-uchebnik/00-276" "field_display_title" => "276" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2088 …2} "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 {#2089 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2090 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2091 …2} "content" => Illuminate\Database\Eloquent\Collection {#2108 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2110 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#2109 #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" => 534473 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_page_end" => null "field_url" => "/5-klass/matematika/bunimovich-uchebnik/00-277" "field_display_title" => "277" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2111 …2} "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 {#2112 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2113 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2114 …2} "content" => Illuminate\Database\Eloquent\Collection {#2131 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2133 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 534473 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_page_end" => null "field_url" => "/5-klass/matematika/bunimovich-uchebnik/00-277" "field_display_title" => "277" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2111 …2} "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 {#2112 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2113 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2114 …2} "content" => Illuminate\Database\Eloquent\Collection {#2131 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2133 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Task {#2132 #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" => 534474 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_page_end" => null "field_url" => "/5-klass/matematika/bunimovich-uchebnik/00-278" "field_display_title" => "278" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2134 …2} "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 {#2135 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2136 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2137 …2} "content" => Illuminate\Database\Eloquent\Collection {#2154 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2156 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 534474 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_page_end" => null "field_url" => "/5-klass/matematika/bunimovich-uchebnik/00-278" "field_display_title" => "278" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2134 …2} "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 {#2135 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2136 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2137 …2} "content" => Illuminate\Database\Eloquent\Collection {#2154 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2156 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 6 => App\Models\Task {#2155 #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" => 534475 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_page_end" => null "field_url" => "/5-klass/matematika/bunimovich-uchebnik/00-279" "field_display_title" => "279" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2157 …2} "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 {#2158 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2159 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2160 …2} "content" => Illuminate\Database\Eloquent\Collection {#2162 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2168 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 534475 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_page_end" => null "field_url" => "/5-klass/matematika/bunimovich-uchebnik/00-279" "field_display_title" => "279" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2157 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null …11 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 949379 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_url" => "/5-klass/matematika/bunimovich-uchebnik/page-85" "field_display_title" => "85" "field_folder" => "1" "field_image_name" => "85" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1147} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1185} "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 {#1183} "content" => Illuminate\Database\Eloquent\Collection {#1194} "next" => array:2 [ "refs" => "949380" "type" => "book_page" ] "previous" => array:2 [ "refs" => "949378" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1844} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 534474 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "85" "field_page_end" => null "field_url" => "/5-klass/matematika/bunimovich-uchebnik/00-278" "field_display_title" => "278" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331} "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 {#1026} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1069} "content" => Illuminate\Database\Eloquent\Collection {#1114} "next" => array:2 [ "refs" => "534475" "type" => "task" ] "previous" => array:2 [ "refs" => "534473" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1131} "page" => array:2 [ "refs" => "949379" "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 => "*" ] }
№278 (с. 85)
Условие. №278 (с. 85)
скриншот условия
278 1) Начертите угол $BOC$. Постройте угол $AOB$, дополняющий его до развёрнутого угла. Постройте угол $DOC$, дополняющий угол $BOC$ до развёрнутого.
2) Каким является угол $AOB$, если угол $BOC$ острый; прямой; тупой?
3) Верно ли, что углы $AOB$ и $DOC$ равны? Почему?
Решение 2. №278 (с. 85)
1)
Чтобы выполнить построение, следуйте шагам:
- Начертите произвольный угол $BOC$. Он имеет вершину в точке $O$ и стороны, образованные лучами $OB$ и $OC$.
- Чтобы построить угол $AOB$, дополняющий угол $BOC$ до развёрнутого, необходимо из вершины $O$ провести луч $OA$ так, чтобы он являлся продолжением луча $OC$. В результате лучи $OA$ и $OC$ образуют прямую линию $AC$. Углы $AOB$ и $BOC$ будут смежными, а их сумма будет равна $180^\circ$.
- Аналогично, чтобы построить угол $DOC$, дополняющий угол $BOC$ до развёрнутого, необходимо из вершины $O$ провести луч $OD$ так, чтобы он являлся продолжением луча $OB$. В результате лучи $OD$ и $OB$ образуют прямую линию $DB$. Углы $DOC$ и $BOC$ также будут смежными, и их сумма будет равна $180^\circ$.
В результате такого построения получатся две прямые, $AC$ и $BD$, пересекающиеся в точке $O$.
Ответ: Построение описано выше. В результате получаются две пересекающиеся в точке $O$ прямые $AC$ и $BD$.
2)
Углы $AOB$ и $BOC$ дополняют друг друга до развёрнутого угла, значит, они являются смежными. Сумма смежных углов равна $180^\circ$.
$\angle AOB + \angle BOC = 180^\circ$
Из этого следует, что $\angle AOB = 180^\circ - \angle BOC$.
Рассмотрим три случая в зависимости от вида угла $BOC$:
- Если угол $BOC$ — острый ($0^\circ < \angle BOC < 90^\circ$), то $\angle AOB = 180^\circ - \angle BOC$ будет больше $90^\circ$ и меньше $180^\circ$. Следовательно, угол $AOB$ — тупой.
- Если угол $BOC$ — прямой ($\angle BOC = 90^\circ$), то $\angle AOB = 180^\circ - 90^\circ = 90^\circ$. Следовательно, угол $AOB$ — также прямой.
- Если угол $BOC$ — тупой ($90^\circ < \angle BOC < 180^\circ$), то $\angle AOB = 180^\circ - \angle BOC$ будет больше $0^\circ$ и меньше $90^\circ$. Следовательно, угол $AOB$ — острый.
Ответ: Если угол $BOC$ острый, то угол $AOB$ — тупой; если угол $BOC$ прямой, то угол $AOB$ — прямой; если угол $BOC$ тупой, то угол $AOB$ — острый.
3)
Да, утверждение, что углы $AOB$ и $DOC$ равны, верно. Вот почему:
По условию, угол $AOB$ дополняет угол $BOC$ до развёрнутого, что математически записывается как:
$\angle AOB + \angle BOC = 180^\circ$
Также по условию, угол $DOC$ дополняет угол $BOC$ до развёрнутого, что записывается как:
$\angle DOC + \angle BOC = 180^\circ$
Мы имеем два выражения, которые равны $180^\circ$. Следовательно, мы можем приравнять их левые части:
$\angle AOB + \angle BOC = \angle DOC + \angle BOC$
Вычтем из обеих частей равенства величину угла $BOC$:
$\angle AOB = \angle DOC$
Это доказывает, что углы $AOB$ и $DOC$ равны. Такие углы также называют вертикальными.
Ответ: Да, верно. Углы $AOB$ и $DOC$ равны, так как каждый из них является дополнением одного и того же угла $BOC$ до $180^\circ$.
Другие задания:
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