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Angle Relationships in Triangles section 4-3 Triangle Sum Theorem Theorem 4-2-1 B The sum of the angle measures of a triangle is 180 . m A+m B+m C A C = 180 An auxiliary line is a line that is added to a figure to aid in a proof. An auxiliary line used in the Triangle Sum Theorem Proof of the Triangle Sum Theorem: Given: ABC Prove: m 1+m 1. Draw line l 2. 1 4; 2+m 3 = 180 1. Parallel Postulate AC through point B. 3 5 2. Alternate Interior Angles Theorem 3. m 4+m 2+m 5 = 180 3. 4. m 1+m 2+m 3 = 180 4. Subst. Prop. Add. Post. & def. of straight problem # 1 - The map of France commonly used in the 1600's was significantly revised as a result of a triangulation land survey. The diagram shows part of the survey map. Use the diagram to find the indicated angle measures. a. m NKM b. m JLK c. m MJK 44 A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem. problem # 2 - a. One of the acute angles of a right triangle measures 23 . What is the measure of the other acute angle? b. The measure of one acute angle of another right triangle is x . Find the measure of the other acute angle. The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure. Exterior Interior An interior angle is formed by two sides of a triangle. An exterior angle is formed by one side of the triangle and the extension of an adjacent side. Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle. 4 is an exterior angle. Its remote interior 's are 1 and 2. problem # 3 - Find m (6x - 1) J. (5x + 17) 126 problem # 4 - Find m C and m J G F H F. F (3y 2 - 72) A B (y2) C D E