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Лемма, страница 8 - гдз по геометрии 10 класс рабочая тетрадь Глазков, Юдина
Авторы: Глазков Ю. А., Юдина И. И., Бутузов В. Ф.
Тип: рабочая тетрадь
Серия: мгу - школе
Издательство: Просвещение
Год издания: 2020 - 2026
Уровень обучения: базовый и углублённый
Цвет обложки: коричневый в сеточку
ISBN: 978-5-09-097573-5
Популярные ГДЗ в 10 классе
1.1. Параллельность прямых, прямой и плоскости - страница 8.
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" => 738475 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "8" "field_page_end" => null "field_url" => "/10-klass/geometrija/glazkov-tetrad/lemma-1" "field_display_title" => "Лемма" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1036 #items: [] #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 {#1034 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1045 #items: array:1 [ 0 => App\Models\Branch {#1044 #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" => 738453 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "1.1. Параллельность прямых, прямой и плоскости" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1046 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "7" "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 {#1056 #items: array:1 [ 0 => App\Models\Book {#1055 #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" => 261 "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 {#1060 #items: array:1 [ 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: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 {#1062 #items: array:1 [ 0 => App\Models\Term {#1059 #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" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #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 {#1064 #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: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 {#1065 #items: array:3 [ 0 => App\Models\Term {#1075 #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" => 4121 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глазков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Юрий" "field_patronymic" => "Александрович" "field_surname_rp" => null ] #original: array:12 [ "id" => 4121 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глазков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Юрий" "field_patronymic" => "Александрович" "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 {#1076 #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" => 6432 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Юдина" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Ирина" "field_patronymic" => "Игоревна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6432 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Юдина" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Ирина" "field_patronymic" => "Игоревна" "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 {#1074 #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" => 3949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бутузов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Валентин" "field_patronymic" => "Фёдорович" "field_surname_rp" => null ] #original: array:12 [ "id" => 3949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бутузов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Валентин" "field_patronymic" => "Фёдорович" "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 {#1071 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1072 #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:10 [ "id" => 6736 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "рабочая тетрадь" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "рабочую тетрадь" "field_creative_case" => "рабочей тетрадью" "field_dative_case" => "рабочей тетради" "field_genitive_case" => "рабочей тетради" "field_nominative_case" => "рабочая тетрадь" "field_prepositional_case" => "рабочей тетради" ] "field_plural_form" => "рабочие тетради" "field_short_name" => "рт" "field_short_name_foreign" => null "field_translit" => "rabochaya tetrad" ] #original: array:10 [ "id" => 6736 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "рабочая тетрадь" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "рабочую тетрадь" "field_creative_case" => "рабочей тетрадью" "field_dative_case" => "рабочей тетради" "field_genitive_case" => "рабочей тетради" "field_nominative_case" => "рабочая тетрадь" "field_prepositional_case" => "рабочей тетради" ] "field_plural_form" => "рабочие тетради" "field_short_name" => "рт" "field_short_name_foreign" => null "field_translit" => "rabochaya tetrad" ] #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 {#1066 #items: array:1 [ 0 => App\Models\Term {#1070 #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 [ "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" => "rossiya" ] #original: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "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" => "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 {#1067 #items: array:1 [ 0 => App\Models\Term {#1068 #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 [ "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" => "moskva" ] #original: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "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" => "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 {#1078 #items: array:1 [ 0 => App\Models\Term {#1069 #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" => 4728 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "мгу - школе" "field_cases" => null "field_foreign_lang_series" => null "field_transcription_eng" => null "field_translit" => "mgu - shkole" ] #original: array:8 [ "id" => 4728 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "мгу - школе" "field_cases" => null "field_foreign_lang_series" => null "field_transcription_eng" => null "field_translit" => "mgu - shkole" ] #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 {#1077 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1081 #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" => 41 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "базовый и углублённый" "field_cases" => array:6 [ "field_accusative_case" => "базовый и углубленный" "field_creative_case" => "базовым и углубленным" "field_dative_case" => "базовому и углубленному" "field_genitive_case" => "базового и углубленного" "field_nominative_case" => "базовый и углубленный" "field_prepositional_case" => "базовом и углубленном" ] "field_translit" => "bazovyy i uglublennyy" ] #original: array:6 [ "id" => 41 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "базовый и углублённый" "field_cases" => array:6 [ "field_accusative_case" => "базовый и углубленный" "field_creative_case" => "базовым и углубленным" "field_dative_case" => "базовому и углубленному" "field_genitive_case" => "базового и углубленного" "field_nominative_case" => "базовый и углубленный" "field_prepositional_case" => "базовом и углубленном" ] "field_translit" => "bazovyy i uglublennyy" ] #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 {#1083 #items: array:1 [ 0 => App\Models\Term {#1080 #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" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "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_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => null ] #original: array:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "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_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => 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_publication_number" => "4" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ 0 => App\Models\Term {#1082 #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" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "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" => "stereotipnoe" ] #original: array:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "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" => "stereotipnoe" ] #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_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "в сеточку" "field_publication_year" => "2020" "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" => "110" "field_priority" => "14" "field_default_folder" => "/geometrija_10/glazkov-rt/" "field_isbn" => "978-5-09-097573-5" "field_cover" => array:1 [ 0 => "/media/geometrija_10/glazkov-rt/covers/cover1.webp?ts=1739295144" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/glazkov-rt/covers/cover1.webp?ts=1739295144" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1079 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/glazkov-tetrad" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Term {#1096 #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" => 51 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "коричневый" "field_cases" => array:6 [ "field_accusative_case" => "коричневый" "field_creative_case" => "коричневым" "field_dative_case" => "коричневому" "field_genitive_case" => "коричневого" "field_nominative_case" => "коричневый" "field_prepositional_case" => "коричневом" ] "field_translit" => "korichnevyy" ] #original: array:6 [ "id" => 51 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "коричневый" "field_cases" => array:6 [ "field_accusative_case" => "коричневый" "field_creative_case" => "коричневым" "field_dative_case" => "коричневому" "field_genitive_case" => "коричневого" "field_nominative_case" => "коричневый" "field_prepositional_case" => "коричневом" ] "field_translit" => "korichnevyy" ] #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" => 381 "subject" => 543 "class_subject" => 392 ] ] #original: array:50 [ "id" => 261 "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 {#1060} "field_class" => Illuminate\Database\Eloquent\Collection {#1062} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1064} "field_author" => Illuminate\Database\Eloquent\Collection {#1065} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1071} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1072} "field_country" => Illuminate\Database\Eloquent\Collection {#1066} "field_city" => Illuminate\Database\Eloquent\Collection {#1067} "field_series" => Illuminate\Database\Eloquent\Collection {#1078} "field_umk" => Illuminate\Database\Eloquent\Collection {#1077} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1081} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1083} "field_publication_number" => "4" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1085} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1084} "field_under_the_edition_degree" => null "field_cover_description" => "в сеточку" "field_publication_year" => "2020" "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" => "110" "field_priority" => "14" "field_default_folder" => "/geometrija_10/glazkov-rt/" "field_isbn" => "978-5-09-097573-5" "field_cover" => array:1 [ 0 => "/media/geometrija_10/glazkov-rt/covers/cover1.webp?ts=1739295144" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/glazkov-rt/covers/cover1.webp?ts=1739295144" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1079} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1086} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1087} "field_url" => "/10-klass/geometrija/glazkov-tetrad" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1097} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #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 {#1048 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 738453 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "1.1. Параллельность прямых, прямой и плоскости" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1046} "field_page_start" => "7" "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 {#1056} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1048} ] #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 {#1037 #items: array:1 [ 0 => App\Models\Branch {#1044} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1041 #items: array:3 [ 0 => App\Models\Element {#1051 #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" => 937184 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1099 #items: array:1 [ 0 => App\Models\Edition {#1089 #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" => 2220 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088 #items: array:1 [ 0 => App\Models\Term {#1090 #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_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 {#1093 #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 {#1094 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 2220 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088} "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 {#1093} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1094} "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" => "738475" "type" => "task" ] "text" => "<p>Лемма. Если одна из двух __________ прямых пересекает данную плоскость, то и __________ эту плоскость.</p><p><strong>Дано: $a \parallel b$, $M$ — точка пересечения прямой $a$ и плоскости $\alpha$.</strong></p><p><strong>Доказать: прямая $b$ __________</strong></p><p><strong>Доказательство.</strong> Пусть $\beta$ — плоскость, в которой лежат параллельные прямые $a$ и $b$. Так как $M \in \alpha$, $M \in \beta$, то __________ плоскости $\alpha$ и $\beta$ пересекаются по некоторой прямой $p$, проходящей через __________ . Таким образом, в плоскости $\beta$ прямая $p$ пересекает прямую $a$ в точке __________ , а потому она __________ и параллельную ей __________ в некоторой точке $N$, причем точка $N \in \alpha$, так как __________ . Итак, $N$ — общая точка прямой __________ и плоскости __________ . Других общих точек с плоскостью $\alpha$ прямая $b$ не имеет. Действительно, если предположить, что прямая $b$ __________ еще одну __________ , прямая $b$ будет целиком лежать в __________ , а значит, будет общей прямой __________ и потому совпадет __________ . Но это невозможно, так как по условию $a \parallel b$, а прямые $a$ и $p$ __________ . Лемма доказана.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "1-1.jpg" "alt" => null "width" => "1677" "height" => 1523 "path" => "/media/geometrija_10/glazkov-rt/0-lemma/1-1.webp?ts=1739375771" ] ] ] #original: array:7 [ "id" => 937184 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1099} "task" => array:2 [ "refs" => "738475" "type" => "task" ] "text" => "<p>Лемма. Если одна из двух __________ прямых пересекает данную плоскость, то и __________ эту плоскость.</p><p><strong>Дано: $a \parallel b$, $M$ — точка пересечения прямой $a$ и плоскости $\alpha$.</strong></p><p><strong>Доказать: прямая $b$ __________</strong></p><p><strong>Доказательство.</strong> Пусть $\beta$ — плоскость, в которой лежат параллельные прямые $a$ и $b$. Так как $M \in \alpha$, $M \in \beta$, то __________ плоскости $\alpha$ и $\beta$ пересекаются по некоторой прямой $p$, проходящей через __________ . Таким образом, в плоскости $\beta$ прямая $p$ пересекает прямую $a$ в точке __________ , а потому она __________ и параллельную ей __________ в некоторой точке $N$, причем точка $N \in \alpha$, так как __________ . Итак, $N$ — общая точка прямой __________ и плоскости __________ . Других общих точек с плоскостью $\alpha$ прямая $b$ не имеет. Действительно, если предположить, что прямая $b$ __________ еще одну __________ , прямая $b$ будет целиком лежать в __________ , а значит, будет общей прямой __________ и потому совпадет __________ . Но это невозможно, так как по условию $a \parallel b$, а прямые $a$ и $p$ __________ . Лемма доказана.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "1-1.jpg" "alt" => null "width" => "1677" "height" => 1523 "path" => "/media/geometrija_10/glazkov-rt/0-lemma/1-1.webp?ts=1739375771" ] ] ] #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 {#1092 #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" => 937294 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:1 [ 0 => App\Models\Edition {#1098 #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" => 2221 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1101 #items: array:1 [ 0 => App\Models\Term {#1100 #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" => 6696 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Skysmart" "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" => "skysmart" ] #original: array:6 [ "id" => 6696 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Skysmart" "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" => "skysmart" ] #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" => "margarita" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1102 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 2221 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1101} "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" => "margarita" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1102} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1103} "field_process_formula" => "" "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" => "738475" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "1-1.png" "alt" => null "width" => "694" "height" => 4395 "path" => "/media/geometrija_10/glazkov-rt/1-lemma/1-1.webp?ts=1739375909" ] ] ] #original: array:6 [ "id" => 937294 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107} "task" => array:2 [ "refs" => "738475" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "1-1.png" "alt" => null "width" => "694" "height" => 4395 "path" => "/media/geometrija_10/glazkov-rt/1-lemma/1-1.webp?ts=1739375909" ] ] ] #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 {#1105 #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" => 1434368 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1115 #items: array:1 [ 0 => App\Models\Edition {#1106 #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" => 5337 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1109 #items: array:1 [ 0 => App\Models\Term {#1108 #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 [ "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" => "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 [ "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" => "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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1110 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1111 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1112 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5337 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1109} "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 {#1110} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1111} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1112} "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" => "738475" "type" => "task" ] "text" => "<p> <strong>Лемма.</strong> Если одна из двух <strong>параллельных</strong> прямых пересекает данную плоскость, то и <strong>другая пересекает</strong> эту плоскость.</p><p> <strong>Дано:</strong> $a \parallel b$, $M$ — точка пересечения прямой $a$ и плоскости $\alpha$.</p><p> <strong>Доказать:</strong> прямая $b$ <strong>пересекает плоскость $\alpha$</strong>.</p><p> <strong>Доказательство.</strong> Пусть $\beta$ — плоскость, в которой лежат параллельные прямые $a$ и $b$. Так как $M \in \alpha$, $M \in \beta$, то плоскости $\alpha$ и $\beta$ пересекаются по некоторой прямой $p$, проходящей через <strong>точку $M$</strong>. Таким образом, в плоскости $\beta$ прямая $p$ пересекает прямую $a$ в точке <strong>$M$</strong>, а потому она <strong>пересекает</strong> и параллельную ей <strong>прямую $b$</strong> в некоторой точке $N$, причем точка $N \in \alpha$, так как <strong>прямая $p$ целиком лежит в плоскости $\alpha$</strong>. Итак, $N$ — общая точка прямой <strong>$b$</strong> и плоскости <strong>$\alpha$</strong>. Других общих точек с плоскостью $\alpha$ прямая $b$ не имеет. Действительно, если предположить, что прямая $b$ <strong>имеет</strong> еще одну <strong>общую точку</strong>, то, согласно <strong>аксиоме (если две точки прямой лежат в плоскости, то вся прямая лежит в этой плоскости)</strong>, прямая $b$ будет целиком лежать в <strong>плоскости $\alpha$</strong>, а значит, будет общей прямой <strong>плоскостей $\alpha$ и $\beta$</strong> и потому совпадет <strong>с прямой $p$</strong>. Но это невозможно, так как по условию $a \parallel b$, а прямые $a$ и $p$ <strong>пересекаются</strong>. Лемма доказана.</p><p> <strong>Ответ:</strong> В пропуски в тексте необходимо вставить следующие слова и словосочетания в указанном порядке:</p><ol> <li>параллельных</li> <li>другая пересекает</li> <li>пересекает плоскость $\alpha$</li> <li>точку $M$</li> <li>$M$</li> <li>пересекает</li> <li>прямую $b$</li> <li>прямая $p$ целиком лежит в плоскости $\alpha$</li> <li>$b$</li> <li>$\alpha$</li> <li>имеет</li> <li>общую точку</li> <li>аксиоме (если две точки прямой лежат в плоскости, то вся прямая лежит в этой плоскости)</li> <li>плоскости $\alpha$</li> <li>плоскостей $\alpha$ и $\beta$</li> <li>с прямой $p$</li> <li>пересекаются</li></ol>" ] #original: array:6 [ "id" => 1434368 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1115} "task" => array:2 [ "refs" => "738475" "type" => "task" ] "text" => "<p> <strong>Лемма.</strong> Если одна из двух <strong>параллельных</strong> прямых пересекает данную плоскость, то и <strong>другая пересекает</strong> эту плоскость.</p><p> <strong>Дано:</strong> $a \parallel b$, $M$ — точка пересечения прямой $a$ и плоскости $\alpha$.</p><p> <strong>Доказать:</strong> прямая $b$ <strong>пересекает плоскость $\alpha$</strong>.</p><p> <strong>Доказательство.</strong> Пусть $\beta$ — плоскость, в которой лежат параллельные прямые $a$ и $b$. Так как $M \in \alpha$, $M \in \beta$, то плоскости $\alpha$ и $\beta$ пересекаются по некоторой прямой $p$, проходящей через <strong>точку $M$</strong>. Таким образом, в плоскости $\beta$ прямая $p$ пересекает прямую $a$ в точке <strong>$M$</strong>, а потому она <strong>пересекает</strong> и параллельную ей <strong>прямую $b$</strong> в некоторой точке $N$, причем точка $N \in \alpha$, так как <strong>прямая $p$ целиком лежит в плоскости $\alpha$</strong>. Итак, $N$ — общая точка прямой <strong>$b$</strong> и плоскости <strong>$\alpha$</strong>. Других общих точек с плоскостью $\alpha$ прямая $b$ не имеет. Действительно, если предположить, что прямая $b$ <strong>имеет</strong> еще одну <strong>общую точку</strong>, то, согласно <strong>аксиоме (если две точки прямой лежат в плоскости, то вся прямая лежит в этой плоскости)</strong>, прямая $b$ будет целиком лежать в <strong>плоскости $\alpha$</strong>, а значит, будет общей прямой <strong>плоскостей $\alpha$ и $\beta$</strong> и потому совпадет <strong>с прямой $p$</strong>. Но это невозможно, так как по условию $a \parallel b$, а прямые $a$ и $p$ <strong>пересекаются</strong>. Лемма доказана.</p><p> <strong>Ответ:</strong> В пропуски в тексте необходимо вставить следующие слова и словосочетания в указанном порядке:</p><ol> <li>параллельных</li> <li>другая пересекает</li> <li>пересекает плоскость $\alpha$</li> <li>точку $M$</li> <li>$M$</li> <li>пересекает</li> <li>прямую $b$</li> <li>прямая $p$ целиком лежит в плоскости $\alpha$</li> <li>$b$</li> <li>$\alpha$</li> <li>имеет</li> <li>общую точку</li> <li>аксиоме (если две точки прямой лежат в плоскости, то вся прямая лежит в этой плоскости)</li> <li>плоскости $\alpha$</li> <li>плоскостей $\alpha$ и $\beta$</li> <li>с прямой $p$</li> <li>пересекаются</li></ol>" ] #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" => "738476" "type" => "task" ] "previous" => array:2 [ "refs" => "738474" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ 0 => App\Models\Book {#1055} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ 0 => App\Models\BookPage {#1120 #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" => 971655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "8" "field_url" => "/10-klass/geometrija/glazkov-tetrad/page-8" "field_display_title" => "8" "field_folder" => "1" "field_image_name" => "8" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1122 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Book {#1055} ] #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 {#1124 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1134 #items: array:1 [ 0 => App\Models\Element {#1133 #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" => 1147771 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Edition {#1089} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "971655" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1951" "height" => 2798 "path" => "/media/geometrija_10/glazkov-rt/0-1/8-1.webp?ts=1745219733" ] ] ] #original: array:6 [ "id" => 1147771 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1135} "book_page" => array:2 [ "refs" => "971655" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1951" "height" => 2798 "path" => "/media/geometrija_10/glazkov-rt/0-1/8-1.webp?ts=1745219733" ] ] ] #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" => "971656" "type" => "book_page" ] "previous" => array:2 [ "refs" => "971654" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1199 #items: array:2 [ 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" => 738474 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "8" "field_page_end" => null "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-7" "field_display_title" => "7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:1 [ 0 => App\Models\Term {#1208 #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 {#1210 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1209 #items: array:1 [ 0 => App\Models\Branch {#1044} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1212 #items: array:1 [ 0 => App\Models\Branch {#1044} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1215 #items: array:3 [ 0 => App\Models\Element {#1224 #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" => 937088 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1225 #items: array:1 [ 0 => App\Models\Edition {#1089} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "738474" "type" => "task" ] "text" => "<p>Вершина $Q$ параллелограмма $MNPQ$ лежит в плоскости $\alpha$, а точки $M$, $N$ и $P$ не лежат в этой плоскости. Докажите, что прямые $NM$ и $NP$ пересекают плоскость $\alpha$.</p><p><strong>Доказательство.</strong></p><p>Прямая $PQ$ пересекает плоскость $\alpha$ в точке $Q$, так как $Q \in \alpha$, поэтому, согласно лемме о пересечении плоскости параллельными прямыми,</p><p>прямая $NM$, параллельная __________, также</p><p>Прямая $MQ$ пересекает __________ , поэтому</p><p>прямая $NP$ __________ , что и требовалось доказать.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "7-1.jpg" "alt" => null "width" => "1677" "height" => 995 "path" => "/media/geometrija_10/glazkov-rt/0-00/7-1.webp?ts=1739375698" ] ] ] #original: array:7 [ "id" => 937088 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1225} "task" => array:2 [ "refs" => "738474" "type" => "task" ] "text" => "<p>Вершина $Q$ параллелограмма $MNPQ$ лежит в плоскости $\alpha$, а точки $M$, $N$ и $P$ не лежат в этой плоскости. Докажите, что прямые $NM$ и $NP$ пересекают плоскость $\alpha$.</p><p><strong>Доказательство.</strong></p><p>Прямая $PQ$ пересекает плоскость $\alpha$ в точке $Q$, так как $Q \in \alpha$, поэтому, согласно лемме о пересечении плоскости параллельными прямыми,</p><p>прямая $NM$, параллельная __________, также</p><p>Прямая $MQ$ пересекает __________ , поэтому</p><p>прямая $NP$ __________ , что и требовалось доказать.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "7-1.jpg" "alt" => null "width" => "1677" "height" => 995 "path" => "/media/geometrija_10/glazkov-rt/0-00/7-1.webp?ts=1739375698" ] ] ] #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 {#1226 #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" => 937198 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1227 #items: array:1 [ 0 => App\Models\Edition {#1098} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "738474" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "7-1.png" "alt" => null "width" => "694" "height" => 1832 "path" => "/media/geometrija_10/glazkov-rt/1-00/7-1.webp?ts=1739375830" ] ] ] #original: array:6 [ "id" => 937198 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1227} "task" => array:2 [ "refs" => "738474" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "7-1.png" "alt" => null "width" => "694" "height" => 1832 "path" => "/media/geometrija_10/glazkov-rt/1-00/7-1.webp?ts=1739375830" ] ] ] #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 {#1228 #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" => 1434272 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1229 #items: array:1 [ 0 => App\Models\Edition {#1106} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "738474" "type" => "task" ] "text" => "<p>Для решения задачи необходимо заполнить пропуски в предложенном доказательстве. Решение основывается на свойствах параллелограмма и лемме о пересечении плоскости параллельными прямыми.</p> <p><strong>Обоснование:</strong></p> <p>1. По свойству параллелограмма $MNPQ$ его противоположные стороны параллельны, то есть $NM \parallel PQ$ и $NP \parallel MQ$.</p> <p>2. Согласно лемме о пересечении плоскости параллельными прямыми: если одна из двух параллельных прямых пересекает плоскость, то и вторая прямая пересекает эту плоскость.</p> <p>3. Прямая $PQ$ пересекает плоскость $\alpha$ в точке $Q$, так как по условию $Q \in \alpha$. Поскольку прямая $NM$ параллельна прямой $PQ$ ($NM \parallel PQ$), то по указанной лемме прямая $NM$ также пересекает плоскость $\alpha$. Это позволяет заполнить первую часть пропусков.</p> <p>4. Аналогично, прямая $MQ$ пересекает плоскость $\alpha$ в точке $Q$ (так как $Q \in \alpha$). Поскольку прямая $NP$ параллельна прямой $MQ$ ($NP \parallel MQ$), то по той же лемме прямая $NP$ также пересекает плоскость $\alpha$. Это позволяет заполнить вторую часть пропусков.</p> <p><strong>Ответ:</strong></p> <p><strong>Доказательство.</strong> Прямая PQ пересекает плоскость $\alpha$ в точке Q, так как $Q \in \alpha$, поэтому, согласно лемме о пересечении плоскости параллельными прямыми, прямая NM, параллельная <strong>прямой PQ</strong>, также <strong>пересекает плоскость $\alpha$</strong>.<br> Прямая MQ пересекает <strong>плоскость $\alpha$ в точке Q</strong>, поэтому прямая NP <strong>также пересекает плоскость $\alpha$</strong>, что и требовалось доказать.</p>" ] #original: array:6 [ "id" => 1434272 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1229} "task" => array:2 [ "refs" => "738474" "type" => "task" ] "text" => "<p>Для решения задачи необходимо заполнить пропуски в предложенном доказательстве. Решение основывается на свойствах параллелограмма и лемме о пересечении плоскости параллельными прямыми.</p> <p><strong>Обоснование:</strong></p> <p>1. По свойству параллелограмма $MNPQ$ его противоположные стороны параллельны, то есть $NM \parallel PQ$ и $NP \parallel MQ$.</p> <p>2. Согласно лемме о пересечении плоскости параллельными прямыми: если одна из двух параллельных прямых пересекает плоскость, то и вторая прямая пересекает эту плоскость.</p> <p>3. Прямая $PQ$ пересекает плоскость $\alpha$ в точке $Q$, так как по условию $Q \in \alpha$. Поскольку прямая $NM$ параллельна прямой $PQ$ ($NM \parallel PQ$), то по указанной лемме прямая $NM$ также пересекает плоскость $\alpha$. Это позволяет заполнить первую часть пропусков.</p> <p>4. Аналогично, прямая $MQ$ пересекает плоскость $\alpha$ в точке $Q$ (так как $Q \in \alpha$). Поскольку прямая $NP$ параллельна прямой $MQ$ ($NP \parallel MQ$), то по той же лемме прямая $NP$ также пересекает плоскость $\alpha$. Это позволяет заполнить вторую часть пропусков.</p> <p><strong>Ответ:</strong></p> <p><strong>Доказательство.</strong> Прямая PQ пересекает плоскость $\alpha$ в точке Q, так как $Q \in \alpha$, поэтому, согласно лемме о пересечении плоскости параллельными прямыми, прямая NM, параллельная <strong>прямой PQ</strong>, также <strong>пересекает плоскость $\alpha$</strong>.<br> Прямая MQ пересекает <strong>плоскость $\alpha$ в точке Q</strong>, поэтому прямая NP <strong>также пересекает плоскость $\alpha$</strong>, что и требовалось доказать.</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" => "738475" "type" => "task" ] "previous" => array:2 [ "refs" => "738473" "type" => "task" ] "book" => 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=> array:2 [ "refs" => "738475" "type" => "task" ] "previous" => array:2 [ "refs" => "738473" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1221} "page" => array:2 [ "refs" => "971655" "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" => 738475 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "8" "field_page_end" => null "field_url" => "/10-klass/geometrija/glazkov-tetrad/lemma-1" 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"book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
Лемма (с. 8)
Условие. Лемма (с. 8)
скриншот условия
Лемма. Если одна из двух __________ прямых пересекает данную плоскость, то и __________ эту плоскость.
Дано: $a \parallel b$, $M$ — точка пересечения прямой $a$ и плоскости $\alpha$.
Доказать: прямая $b$ __________
Доказательство. Пусть $\beta$ — плоскость, в которой лежат параллельные прямые $a$ и $b$. Так как $M \in \alpha$, $M \in \beta$, то __________ плоскости $\alpha$ и $\beta$ пересекаются по некоторой прямой $p$, проходящей через __________ . Таким образом, в плоскости $\beta$ прямая $p$ пересекает прямую $a$ в точке __________ , а потому она __________ и параллельную ей __________ в некоторой точке $N$, причем точка $N \in \alpha$, так как __________ . Итак, $N$ — общая точка прямой __________ и плоскости __________ . Других общих точек с плоскостью $\alpha$ прямая $b$ не имеет. Действительно, если предположить, что прямая $b$ __________ еще одну __________ , прямая $b$ будет целиком лежать в __________ , а значит, будет общей прямой __________ и потому совпадет __________ . Но это невозможно, так как по условию $a \parallel b$, а прямые $a$ и $p$ __________ . Лемма доказана.
Решение. Лемма (с. 8)
Решение 2. Лемма (с. 8)
Лемма. Если одна из двух параллельных прямых пересекает данную плоскость, то и другая пересекает эту плоскость.
Дано: $a \parallel b$, $M$ — точка пересечения прямой $a$ и плоскости $\alpha$.
Доказать: прямая $b$ пересекает плоскость $\alpha$.
Доказательство. Пусть $\beta$ — плоскость, в которой лежат параллельные прямые $a$ и $b$. Так как $M \in \alpha$, $M \in \beta$, то плоскости $\alpha$ и $\beta$ пересекаются по некоторой прямой $p$, проходящей через точку $M$. Таким образом, в плоскости $\beta$ прямая $p$ пересекает прямую $a$ в точке $M$, а потому она пересекает и параллельную ей прямую $b$ в некоторой точке $N$, причем точка $N \in \alpha$, так как прямая $p$ целиком лежит в плоскости $\alpha$. Итак, $N$ — общая точка прямой $b$ и плоскости $\alpha$. Других общих точек с плоскостью $\alpha$ прямая $b$ не имеет. Действительно, если предположить, что прямая $b$ имеет еще одну общую точку, то, согласно аксиоме (если две точки прямой лежат в плоскости, то вся прямая лежит в этой плоскости), прямая $b$ будет целиком лежать в плоскости $\alpha$, а значит, будет общей прямой плоскостей $\alpha$ и $\beta$ и потому совпадет с прямой $p$. Но это невозможно, так как по условию $a \parallel b$, а прямые $a$ и $p$ пересекаются. Лемма доказана.
Ответ: В пропуски в тексте необходимо вставить следующие слова и словосочетания в указанном порядке:
- параллельных
- другая пересекает
- пересекает плоскость $\alpha$
- точку $M$
- $M$
- пересекает
- прямую $b$
- прямая $p$ целиком лежит в плоскости $\alpha$
- $b$
- $\alpha$
- имеет
- общую точку
- аксиоме (если две точки прямой лежат в плоскости, то вся прямая лежит в этой плоскости)
- плоскости $\alpha$
- плоскостей $\alpha$ и $\beta$
- с прямой $p$
- пересекаются
Другие задания:
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