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Proof of Triangle Angle Sum and Exterior Angle Theorem

This document provides a step-by-step geometric proof that the sum of the angles in a triangle equals 180 degrees. It begins by drawing a line parallel to one side of the triangle and using alternate angles to establish the relationship between the angles. It further demonstrates that the exterior angle of a triangle is equal to the sum of the two opposite interior angles. Both proofs utilize fundamental properties of angles formed by parallel lines and triangles, highlighting the interconnectedness of these geometric concepts.

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Proof of Triangle Angle Sum and Exterior Angle Theorem

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  1. Geometric Proof Prove that the sum of the angles in a triangle is 180˚. x x x y y y a a Draw a line parallel to one side. Let x and y be the other two angles formed with the line. b b c c Then x = b (alternate angles) and y = c (alternate angles) We can also see that x + y + a = 180˚. (angles on a line) Therefore, a + b + c = 180˚.

  2. Prove that the exterior angle of a triangle is equal to the sum of the two opposite interior angles. Let the exterior angle be x. a We must show that a + b = x. Let the other interior angle be c. x b c We know that a + b + c = 180˚. (angles in a triangle) We also know that x + c = 180˚. (angles on a straight line) Therefore, a + b = x.

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