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UNIT 4: TRIANGLE CONGRUENCE 4.2 Angle Relationships in Triangles Warm Up 11/16/11 (HW #19 [4.2] Pg 228 #s 15 – 21, 24) 1. Find the measure of exterior DBA of mD = 80°. BCD, if mDBC = 30°, mC= 70°, and 150° 2. What is the complement of an angle with measure 17°? 73° 3. How many lines can be drawn through N parallel to MP? Why? 1; Parallel Post. Objectives (see page 223) Find the measures of interior and exterior angles of triangles. Apply theorems about the interior and exterior angles of triangles. *Standard 12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems. *Standard 13.0 Students prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles. *see page 223 Example 1A: Application (see page 224) 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, and mYWZ. mXYZ + mYZX + mZXY = 180° mXYZ + 40 + 62 = 180 Sum. Thm 118° Substitute 40 for mYZX and 62 for mZXY. mXYZ + 102 = 180 Simplify. Subtract 102 from both sides. mXYZ = 78° mYWX + mWXY + mXYW = 180° mYWX + 118 + 12 = 180 mYWX + 130 = 180 mYWX = 50° Sum. Thm Substitute 118 for mWXY and 12 for mXYW. Simplify. Subtract 130 from both sides. Check It Out! Example 1 (see page 224) Use the diagram to find mMJK. mMJK + mJKM + mKMJ = 180° mMJK + 104 + 44= 180 mMJK + 148 = 180 mMJK = 32° Sum. Thm Substitute 104 for mJKM and 44 for mKMJ. Simplify. Subtract 148 from both sides. *see page 224 A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem. Example 2: Finding Angle Measures in Right Triangles (see page 225) 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 mB = (90 – 2x)° Acute angles of rt. are complementary Substitute 2x for mA. Subtract 2x from both sides. Check It Out! Example 2a (see page 225) 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° 63.7 + mB = 90 mB = 26.3° Acute angles of rt. are complementary Substitute 63.7 for mA. Subtract 63.7 from both sides. Check It Out! Example 2b The measure of one of the acute angles in a right triangle is x°. What is 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. 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. Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle. Exterior Interior 3 is an interior angle. 4 is an exterior angle. The remote interior angles of 4 are 1 and 2. Example 3: Applying the Exterior Angle Theorem Find mB. Check It Out! Example 3 Find mACD. Example 4: Applying the Third Angles Theorem Find mK and mJ. Check It Out! Example 4 Find mP and mT. Lesson Quiz: Part I 1. The measure of one of the acute angles in a right triangle is 56°. What is the measure of the other acute angle? 2. Find mABD. 3. Find mN and mP. Lesson Quiz: Part II 4. The diagram is a map showing John's house, Kay's house, and the grocery store. What is the angle the two houses make with the store?