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4-2 Angle Relationships in Triangles Toolbox Pg. 228(15-24;42 why4; 53) Holt McDougal Geometry 4-2 Angle Relationships in Triangles Essential Questions How do you find the measures of interior and exterior angles of triangles? How do you apply theorems about the interior and exterior angles of triangles? Holt McDougal Geometry 4-2 Angle Relationships in Triangles Holt McDougal Geometry 4-2 Angle Relationships in Triangles Example 1: Application After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find mXYZ. mXYZ + mYZX + mZXY = 180° mXYZ + 40 + 62 = 180 mXYZ + 102 = 180 mXYZ = 78° Holt McDougal Geometry Sum. Thm Substitute 40 for mYZX and 62 for mZXY. Simplify. Subtract 102 from both sides. 4-2 Angle Relationships in Triangles A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem. Holt McDougal Geometry 4-2 Angle Relationships in Triangles Example 2: Finding Angle Measures in Right Triangles One of the acute angles in a right triangle measures 2x°. What is the measure of the other acute angle? Let the acute angles be A and B, with mA = 2x°. mA + mB = 90° 2x + mB = 90 Acute s of rt. are comp. Substitute 2x for mA. mB = (90 – 2x)° Subtract 2x from both sides. Holt McDougal Geometry 4-2 Angle Relationships in Triangles The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure. Exterior Interior Holt McDougal Geometry 4-2 Angle Relationships in Triangles An interior angle is formed by two sides of a triangle. An exterior angle is formed by one side of the triangle and extension of an adjacent side. 4 is an exterior angle. Exterior Interior 3 is an interior angle. Holt McDougal Geometry 4-2 Angle Relationships in Triangles 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. Exterior Interior The remote interior angles of 4 are 1 and 2. 3 is an interior angle. Holt McDougal Geometry 4-2 Angle Relationships in Triangles Holt McDougal Geometry 4-2 Angle Relationships in Triangles Example 3: Applying the Exterior Angle Theorem Find mB. mA + mB = mBCD 15 + 2x + 3 = 5x – 60 2x + 18 = 5x – 60 78 = 3x Ext. Thm. Substitute 15 for mA, 2x + 3 for mB, and 5x – 60 for mBCD. Simplify. Subtract 2x and add 60 to both sides. Divide by 3. 26 = x mB = 2x + 3 = 2(26) + 3 = 55° Holt McDougal Geometry 4-2 Angle Relationships in Triangles Holt McDougal Geometry 4-2 Angle Relationships in Triangles Example 4: Applying the Third Angles Theorem Find mK and mJ. K J mK = mJ Third s Thm. Def. of s. 4y2 = 6y2 – 40 Substitute 4y2 for mK and 6y2 – 40 for mJ. –2y2 = –40 y2 = 20 Subtract 6y2 from both sides. Divide both sides by -2. So mK = 4y2 = 4(20) = 80°. Since mJ = mK, mJ = 80°. Holt McDougal Geometry