@prefix rdf: . @prefix dbr: . @prefix yago: . dbr:Exterior_angle_theorem rdf:type yago:WikicatTheoremsInPlaneGeometry , yago:Statement106722453 , yago:Communication100033020 , yago:Theorem106752293 , yago:Proposition106750804 , yago:Message106598915 , yago:Abstraction100002137 . @prefix rdfs: . dbr:Exterior_angle_theorem rdfs:label "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u043E \u0432\u043D\u0435\u0448\u043D\u0435\u043C \u0443\u0433\u043B\u0435 \u0442\u0440\u0435\u0443\u0433\u043E\u043B\u044C\u043D\u0438\u043A\u0430"@ru , "Primo teorema dell'angolo esterno"@it , "\u5916\u89D2\u5B9A\u7406"@ja , "Teorema del \u00E1ngulo exterior"@es , "Exterior angle theorem"@en , "\u5916\u89D2\u5B9A\u7406"@zh , "Teorema dos \u00E2ngulos externos"@pt , "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u043F\u0440\u043E \u0437\u043E\u0432\u043D\u0456\u0448\u043D\u0456\u0439 \u043A\u0443\u0442 \u0442\u0440\u0438\u043A\u0443\u0442\u043D\u0438\u043A\u0430"@uk , "Au\u00DFenwinkelsatz"@de ; rdfs:comment "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u043E \u0432\u043D\u0435\u0448\u043D\u0435\u043C \u0443\u0433\u043B\u0435 \u0442\u0440\u0435\u0443\u0433\u043E\u043B\u044C\u043D\u0438\u043A\u0430 \u2014 \u043E\u0434\u043D\u0430 \u0438\u0437 \u043E\u0441\u043D\u043E\u0432\u043D\u044B\u0445 \u0442\u0435\u043E\u0440\u0435\u043C \u043F\u043B\u0430\u043D\u0438\u043C\u0435\u0442\u0440\u0438\u0438."@ru , "O teorema dos \u00E2ngulos externos de um tri\u00E2ngulo \u00E9 um teorema de geometria que diz que o \u00E2ngulo externo de um tri\u00E2ngulo \u00E9 maior que os dois \u00E2ngulos internos n\u00E3o adjacentes a ele ou ainda que o \u00E2ngulo externo de um tri\u00E2ngulo \u00E9 igual \u00E0 soma dos dois \u00E2ngulos internos n\u00E3o adjacentes a ele. Um tri\u00E2ngulo tem tr\u00EAs v\u00E9rtices. Os lados de um tri\u00E2ngulo que se em um v\u00E9rtice e formam um \u00E2ngulo, que \u00E9 chamado de \u00E2ngulo interno."@pt , "El teorema del \u00E1ngulo exterior es la Proposici\u00F3n 1.16 en los Elementos de Euclides que dice lo siguiente: = mABC + mBAC (aqu\u00ED, mACD denota la medida del \u00E1ngulo ACD) Prueba:"@es , "\u5916\u89D2\u5B9A\u7406\uFF08\u304C\u3044\u304B\u304F\u3066\u3044\u308A\uFF09\u3068\u306F\u3001\u4E09\u89D2\u5F62\u306E\u5916\u89D2\u306F\u305D\u308C\u3068\u96A3\u308A\u5408\u308F\u306A\u30442\u3064\u306E\u5185\u89D2\u306E\u548C\u306B\u7B49\u3057\u3044\u3068\u3044\u3046\u3053\u3068\u3092\u793A\u3059\u3001\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u5E7E\u4F55\u5B66\u306B\u304A\u3051\u308B\u5B9A\u7406\u3002\u305D\u306E\u5F62\u72B6\u304B\u3089\u3001\u300C\u30B9\u30EA\u30C3\u30D1\u306E\u6CD5\u5247\u300D\u3068\u547C\u3070\u308C\u308B\u3053\u3068\u3082\u3042\u308B\u3002"@ja , "The exterior angle theorem is Proposition 1.16 in Euclid's Elements, which states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. This is a fundamental result in absolute geometry because its proof does not depend upon the parallel postulate. Some authors refer to the \"High school exterior angle theorem\" as the strong form of the exterior angle theorem and \"Euclid's exterior angle theorem\" as the weak form."@en , "\u5916\u89D2\u5B9A\u7406\uFF0C\u901A\u5E38\u662F\u6307\u4E09\u89D2\u5F62\u4E2D\uFF0C\u4EFB\u4E00\u89D2\u7684\u5916\u89D2\uFF0C\u7B49\u65BC\u53E6\u5169\u89D2\u7684\u548C\u3002\u5916\u89D2\u5B9A\u7406\u4E5F\u53EF\u4EE5\u64F4\u5145\u5230\u4EFB\u610F\u591A\u908A\u5F62\u4E2D\uFF1A\u4EFB\u610F\u591A\u908A\u5F62\u7684\u5916\u89D2\u548C\uFF0C\u7B49\u65BC\u4E00\u5468\u89D2\u3002"@zh , "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u043F\u0440\u043E \u0437\u043E\u0432\u043D\u0456\u0448\u043D\u0456\u0439 \u043A\u0443\u0442 \u0442\u0440\u0438\u043A\u0443\u0442\u043D\u0438\u043A\u0430 \u2014 \u0446\u0435 \u0442\u0432\u0435\u0440\u0434\u0436\u0435\u043D\u043D\u044F \u043F\u0440\u043E \u0432\u043B\u0430\u0441\u0442\u0438\u0432\u0456\u0441\u0442\u044C \u0437\u043E\u0432\u043D\u0456\u0448\u043D\u044C\u043E\u0433\u043E \u043A\u0443\u0442\u0430 \u0442\u0440\u0438\u043A\u0443\u0442\u043D\u0438\u043A\u0430, \u0437\u0430 \u044F\u043A\u0438\u043C \u0437\u043E\u0432\u043D\u0456\u0448\u043D\u0456\u0439 \u043A\u0443\u0442 \u0442\u0440\u0438\u043A\u0443\u0442\u043D\u0438\u043A\u0430 \u0434\u043E\u0440\u0456\u0432\u043D\u044E\u0454 \u0441\u0443\u043C\u0456 \u0434\u0432\u043E\u0445 \u0432\u043D\u0443\u0442\u0440\u0456\u0448\u043D\u0456\u0445 \u043A\u0443\u0442\u0456\u0432, \u043D\u0435 \u0441\u0443\u043C\u0456\u0436\u043D\u0438\u0445 \u0456\u0437 \u043D\u0438\u043C."@uk , "Der Au\u00DFenwinkelsatz (englisch Exterior Angle Theorem) ist ein Lehrsatz der Geometrie, der besagt, dass jeder Au\u00DFenwinkel eines Dreiecks so gro\u00DF ist wie die beiden nicht anliegenden Innenwinkel zusammen. Er wurde erstmals im 3. Jh. v. Chr. als Satz 32 in Buch 1 der Elemente Euklids bewiesen."@de , "Il primo teorema dell'angolo esterno \u00E8 uno dei principali teoremi della geometria euclidea."@it . @prefix foaf: . dbr:Exterior_angle_theorem foaf:depiction , , . @prefix dcterms: . @prefix dbc: . dbr:Exterior_angle_theorem dcterms:subject dbc:Articles_containing_proofs , dbc:Theorems_about_triangles , dbc:Angle . @prefix dbo: . dbr:Exterior_angle_theorem dbo:wikiPageID 19298354 ; dbo:wikiPageRevisionID 1122191442 ; dbo:wikiPageWikiLink , dbr:India , dbr:Parallel_postulate , dbr:Triangle , dbr:North_Pole , , dbr:Compass-and-straightedge_construction , , dbr:Hyperbolic_geometry , dbr:Equator , , , dbr:Maharashtra , dbr:Exterior_angle , dbr:Spherical_geometry , dbc:Theorems_about_triangles , , dbr:Sum_of_angles_of_a_triangle , dbc:Articles_containing_proofs , dbr:Absolute_geometry , dbr:United_States , dbr:Elliptical_geometry , dbr:Foundations_of_geometry , dbr:Logically_equivalent , dbc:Angle , dbr:Spherical_triangle , dbr:Great_circle . @prefix owl: . @prefix wikidata: . dbr:Exterior_angle_theorem owl:sameAs wikidata:Q5421989 , , , , . @prefix yago-res: . dbr:Exterior_angle_theorem owl:sameAs yago-res:Exterior_angle_theorem , , , , , , , , , , . @prefix dbp: . @prefix dbt: . dbr:Exterior_angle_theorem dbp:wikiPageUsesTemplate dbt:Clear , dbt:Reflist , dbt:ISBN , dbt:Citation , dbt:Short_description , dbt:Ancient_Greek_mathematics , dbt:Cite_book ; dbo:thumbnail ; dbo:abstract "Il primo teorema dell'angolo esterno \u00E8 uno dei principali teoremi della geometria euclidea."@it , "Der Au\u00DFenwinkelsatz (englisch Exterior Angle Theorem) ist ein Lehrsatz der Geometrie, der besagt, dass jeder Au\u00DFenwinkel eines Dreiecks so gro\u00DF ist wie die beiden nicht anliegenden Innenwinkel zusammen. Er wurde erstmals im 3. Jh. v. Chr. als Satz 32 in Buch 1 der Elemente Euklids bewiesen."@de , "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u043E \u0432\u043D\u0435\u0448\u043D\u0435\u043C \u0443\u0433\u043B\u0435 \u0442\u0440\u0435\u0443\u0433\u043E\u043B\u044C\u043D\u0438\u043A\u0430 \u2014 \u043E\u0434\u043D\u0430 \u0438\u0437 \u043E\u0441\u043D\u043E\u0432\u043D\u044B\u0445 \u0442\u0435\u043E\u0440\u0435\u043C \u043F\u043B\u0430\u043D\u0438\u043C\u0435\u0442\u0440\u0438\u0438."@ru , "\u5916\u89D2\u5B9A\u7406\uFF0C\u901A\u5E38\u662F\u6307\u4E09\u89D2\u5F62\u4E2D\uFF0C\u4EFB\u4E00\u89D2\u7684\u5916\u89D2\uFF0C\u7B49\u65BC\u53E6\u5169\u89D2\u7684\u548C\u3002\u5916\u89D2\u5B9A\u7406\u4E5F\u53EF\u4EE5\u64F4\u5145\u5230\u4EFB\u610F\u591A\u908A\u5F62\u4E2D\uFF1A\u4EFB\u610F\u591A\u908A\u5F62\u7684\u5916\u89D2\u548C\uFF0C\u7B49\u65BC\u4E00\u5468\u89D2\u3002"@zh , "O teorema dos \u00E2ngulos externos de um tri\u00E2ngulo \u00E9 um teorema de geometria que diz que o \u00E2ngulo externo de um tri\u00E2ngulo \u00E9 maior que os dois \u00E2ngulos internos n\u00E3o adjacentes a ele ou ainda que o \u00E2ngulo externo de um tri\u00E2ngulo \u00E9 igual \u00E0 soma dos dois \u00E2ngulos internos n\u00E3o adjacentes a ele. Um tri\u00E2ngulo tem tr\u00EAs v\u00E9rtices. Os lados de um tri\u00E2ngulo que se em um v\u00E9rtice e formam um \u00E2ngulo, que \u00E9 chamado de \u00E2ngulo interno. Na figura abaixo observamos que os \u00E2ngulos , , s\u00E3o os \u00E2ngulos internos do tri\u00E2ngulo e temos que \u00E9 um \u00E2ngulo externo. Assim, um \u00E2ngulo externo de um tri\u00E2ngulo \u00E9 o \u00E2ngulo formado pelo prolongamento de um lado e o lado adjacente. O \u00E2ngulo externo \u00E9 suplementar ao interno adjacente. Na verdade existem dois teoremas do \u00E2ngulo externo, isto \u00E9, dois resultados que associam o \u00E2ngulo externo aos \u00E2ngulos internos n\u00E3o adjacentes, por\u00E9m ambos se complementam e podem ser vistos como um \u00FAnico teorema. S\u00E3o eles: 1. \n* Em todo tri\u00E2ngulo, qualquer \u00E2ngulo externo \u00E9 maior que qualquer um dos \u00E2ngulos internos n\u00E3o adjacentes. 2. \n* Em todo tri\u00E2ngulo, qualquer \u00E2ngulo externo \u00E9 igual a soma dos dois \u00E2ngulos internos n\u00E3o adjacentes."@pt , "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u043F\u0440\u043E \u0437\u043E\u0432\u043D\u0456\u0448\u043D\u0456\u0439 \u043A\u0443\u0442 \u0442\u0440\u0438\u043A\u0443\u0442\u043D\u0438\u043A\u0430 \u2014 \u0446\u0435 \u0442\u0432\u0435\u0440\u0434\u0436\u0435\u043D\u043D\u044F \u043F\u0440\u043E \u0432\u043B\u0430\u0441\u0442\u0438\u0432\u0456\u0441\u0442\u044C \u0437\u043E\u0432\u043D\u0456\u0448\u043D\u044C\u043E\u0433\u043E \u043A\u0443\u0442\u0430 \u0442\u0440\u0438\u043A\u0443\u0442\u043D\u0438\u043A\u0430, \u0437\u0430 \u044F\u043A\u0438\u043C \u0437\u043E\u0432\u043D\u0456\u0448\u043D\u0456\u0439 \u043A\u0443\u0442 \u0442\u0440\u0438\u043A\u0443\u0442\u043D\u0438\u043A\u0430 \u0434\u043E\u0440\u0456\u0432\u043D\u044E\u0454 \u0441\u0443\u043C\u0456 \u0434\u0432\u043E\u0445 \u0432\u043D\u0443\u0442\u0440\u0456\u0448\u043D\u0456\u0445 \u043A\u0443\u0442\u0456\u0432, \u043D\u0435 \u0441\u0443\u043C\u0456\u0436\u043D\u0438\u0445 \u0456\u0437 \u043D\u0438\u043C."@uk , "The exterior angle theorem is Proposition 1.16 in Euclid's Elements, which states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. This is a fundamental result in absolute geometry because its proof does not depend upon the parallel postulate. In several high school treatments of geometry, the term \"exterior angle theorem\" has been applied to a different result, namely the portion of Proposition 1.32 which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. This result, which depends upon Euclid's parallel postulate will be referred to as the \"High school exterior angle theorem\" (HSEAT) to distinguish it from Euclid's exterior angle theorem. Some authors refer to the \"High school exterior angle theorem\" as the strong form of the exterior angle theorem and \"Euclid's exterior angle theorem\" as the weak form."@en , "\u5916\u89D2\u5B9A\u7406\uFF08\u304C\u3044\u304B\u304F\u3066\u3044\u308A\uFF09\u3068\u306F\u3001\u4E09\u89D2\u5F62\u306E\u5916\u89D2\u306F\u305D\u308C\u3068\u96A3\u308A\u5408\u308F\u306A\u30442\u3064\u306E\u5185\u89D2\u306E\u548C\u306B\u7B49\u3057\u3044\u3068\u3044\u3046\u3053\u3068\u3092\u793A\u3059\u3001\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u5E7E\u4F55\u5B66\u306B\u304A\u3051\u308B\u5B9A\u7406\u3002\u305D\u306E\u5F62\u72B6\u304B\u3089\u3001\u300C\u30B9\u30EA\u30C3\u30D1\u306E\u6CD5\u5247\u300D\u3068\u547C\u3070\u308C\u308B\u3053\u3068\u3082\u3042\u308B\u3002"@ja , "El teorema del \u00E1ngulo exterior es la Proposici\u00F3n 1.16 en los Elementos de Euclides que dice lo siguiente: = mABC + mBAC (aqu\u00ED, mACD denota la medida del \u00E1ngulo ACD) Prueba:"@es . @prefix gold: . dbr:Exterior_angle_theorem gold:hypernym dbr:Proposition . @prefix prov: . dbr:Exterior_angle_theorem prov:wasDerivedFrom . @prefix xsd: . dbr:Exterior_angle_theorem dbo:wikiPageLength "8108"^^xsd:nonNegativeInteger . @prefix wikipedia-en: . dbr:Exterior_angle_theorem foaf:isPrimaryTopicOf wikipedia-en:Exterior_angle_theorem .