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Angle Relationships in Triangles First 10! Step 1: Draw a triangle and label the vertices ABC. Step 2: Measure each angle with a protractor. Step 3: Calculate the sum of the angle measures. Step 4: Did your neighbor find the same thing? Holt McDougal Geometry Angle Relationships in Triangles Holt McDougal Geometry 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 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. 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 mYWZ. 118° Step 1 Find mWXY. mYXZ + mWXY = 180° 62 + mWXY = 180 mWXY = 118° Holt McDougal Geometry Lin. Pair Thm. and Add. Post. Substitute 62 for mYXZ. Subtract 62 from both sides. Angle Relationships in Triangles Example 1: Application Continued 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 mYWZ. 118° Step 2 Find mYWZ. mYWX + mWXY + mXYW = 180° Sum. Thm mYWX + 118 + 12 = 180 Substitute 118 for mWXY and 12 for mXYW. mYWX + 130 = 180 Simplify. mYWX = 50° Subtract 130 from both sides. Holt McDougal Geometry Angle Relationships in Triangles Example 2 The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of the other acute angle? Let the acute angles be A and B, with mA = 63.7°. mA + mB = 90° Acute s of rt. 63.7 + mB = 90 Substitute 63.7 for mA. mB = 26.3° Holt McDougal Geometry are comp. Subtract 63.7 from both sides. 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. It is formed by one side of the triangle and extension of an adjacent side. Exterior Interior Holt McDougal Geometry 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 Angle Relationships in Triangles Example 3: Applying the Exterior Angle Theorem Find mB. mA + mB = mBCD Ext. Thm. 15 + 2x + 3 = 5x – 60 Substitute 15 for mA, 2x + 3 for mB, and 5x – 60 for mBCD. 2x + 18 = 5x – 60 78 = 3x Simplify. Subtract 2x and add 60 to both sides. Divide by 3. 26 = x mB = 2x + 3 = 2(26) + 3 = 55° Holt McDougal Geometry Angle Relationships in Triangles Example 3 Find mACD. mACD = mA + mB Ext. Thm. 6z – 9 = 2z + 1 + 90 Substitute 6z – 9 for mACD, 2z + 1 for mA, and 90 for mB. 6z – 9 = 2z + 91 Simplify. 4z = 100 Subtract 2z and add 9 to both sides. Divide by 4. z = 25 mACD = 6z – 9 = 6(25) – 9 = 141° Holt McDougal Geometry Angle Relationships in Triangles Example 4 Find mP and mT. P T mP = mT Third s Thm. Def. of s. 2x2 = 4x2 – 32 Substitute 2x2 for mP and 4x2 – 32 for mT. –2x2 = –32 x2 = 16 Subtract 4x2 from both sides. Divide both sides by -2. So mP = 2x2 = 2(16) = 32°. Since mP = mT, mT = 32°. Holt McDougal Geometry Angle Relationships in Triangles Geometric figures are congruent if they are the same size and shape. Corresponding angles and corresponding sides are in the same position in polygons with an equal number of sides. Two polygons are congruent polygons if and only if their corresponding sides are congruent. Thus triangles that are the same size and shape are congruent. Holt McDougal Geometry Angle Relationships in Triangles Holt McDougal Geometry Angle Relationships in Triangles Example 1 If polygon LMNP polygon EFGH, identify all pairs of corresponding congruent parts. Angles: L E, M F, N G, P H Sides: LM EF, MN FG, NP GH, LP EH Holt McDougal Geometry Angle Relationships in Triangles Example 2a Given: ∆ABC ∆DEF a.) Find the value of x. b.) Find mF. AB DE Corr. sides of ∆s are . AB = DE Def. of parts. 2x – 2 = 6 2x = 8 x=4 Holt McDougal Geometry Substitute values for AB and DE. Add 2 to both sides. Divide both sides by 2.