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exterior angle theorem
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''.
==Exterior angles==
A triangle has three corners, called ''vertices''. The sides of a triangle (line segments) that come together at a vertex form two angles (four angles if you consider the sides of the triangle to be lines instead of line segments).〔One line segment is considered the initial side and the other the terminal side. The angle is formed by going counterclockwise from the initial side to the terminal side. The choice of which line segment is the initial side is arbitrary, so there are two possibilities for the angle determined by the line segments.〕 Only one of these angles contains the third side of the triangle in its interior, and this angle is called an ''interior angle'' of the triangle.〔This way of defining interior angles does not presuppose that the sum of the angles of a triangle is 180 degrees.〕 In the picture below, the angles ''∠ABC'', ''∠BCA'' and ''∠CAB'' are the three interior angles of the triangle. An ''exterior angle'' is formed by extending one of the sides of the triangle; the angle between the extended side and the other side is the exterior angle. In the picture, angle ''∠ACD'' is an exterior angle.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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