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Angles of Triangles Section 4.2 Objectives Find angle measures in triangles. Key Vocabulary Corollary Exterior angles Interior angles Measures of Angles of a Triangle The word “triangle” means “three angles” When the sides of a triangles are extended, however, other angles are formed The original 3 angles of the triangle are the interior angles The angles that are adjacent to interior angles are the exterior angles Each vertex has a pair of exterior angles Original Triangle Extend sides Exterior Angle Exterior Angle Interior Angle Triangle Interior and Exterior Angles Smiley faces are interior angles and hearts represent the exterior angles B A C Triangle Interior and Exterior Angles A ))) Interior Angles C B ( D Exterior Angles (formed by extending the sides) E F Triangle Sum Theorem The Triangle Angle-Sum Theorem gives the relationship among the interior angle measures of any triangle. Theorem 4.1 – Triangle Sum Theorem The sum of the measures of the angles of a triangle is 180°. X mX + mY + mZ = 180° Y Z Triangle Sum Theorem Example 1 Given mA = 43° and mB = 85°, find mC. SOLUTION mA + mB + mC = 180° 43° + 85° + mC = 180° 128° + mC = 180° 128° + mC – 128° = 180° – 128° mC = 52° ANSWER CHECK Triangle Sum Theorem Substitute 43° for mA and 85° for mB. Simplify. Subtract 128° from each side. Simplify. C has a measure of 52°. Check your solution by substituting 52° for mC. 43° + 85° + 52° = 180° Example 2a A. Find p in the acute triangle. 73° + 44° + p° = 180° 117 + p = 180 Triangle Sum Theorem Example 2b B. Find m in the obtuse triangle. 62 23° + 62° + m° = 180° Triangle Sum Theorem 23 m Your Turn: A. Find a in the acute triangle. 88° + 38° + a° = 180° Triangle Sum Theorem 38° a° 88° Your Turn: B. Find c in the obtuse triangle. 24° + 38° + c° = 180° Triangle Sum Theorem. 38° 24° c° Properties of Triangles Now do these: 41o 80o 30o b= c c= 79o y z x 34o 62o 54o a a= 141o b q 57o 58o x = 180 – 141 = 39 y = 180 – (58+39) = 83 z = 180 – 83 = 97 p p = 180 – (90+57) = 33 r q = 57 (vertically opposite angles are equal) r = 180 – (79+57) = 44 Example 3 Find the angle measures in the scalene triangle. 2x° + 3x° + 5x° = 180° 10x = 180 10 10 Triangle Sum Theorem Simplify. Divide both sides by 10. x = 18 The angle labeled 2x° measures 2(18°) = 36°, the angle labeled 3x° measures 3(18°) = 54°, and the angle labeled 5x° measures 5(18°) = 90°. Your Turn: Find the angle measures in the scalene triangle. 3x° + 7x° + 10x° = 180° Triangle Sum Theorem The angle labeled 3x° measures _____ = __°, the angle labeled 7x° measures ____= ___°, and the angle labeled 10x° measures _____ = ___°. 10x° 3x° 7x° Example 4: Find the missing angle measures. Find first because the measure of two angles of the triangle are known. Angle Sum Theorem Simplify. Subtract 117 from each side. Example 4: Angle Sum Theorem Simplify. Subtract 142 from each side. Answer: Your Turn: Find the missing angle measures. Answer: Triangle Angle-Sum Corollaries Corollary 4.1 – The acute s of a right ∆ are complementary. Example: m∠x + m∠y = 90˚ x° y° Example 5 ∆ABC and ∆ABD are right triangles. Suppose mABD = 35°. a. Find mDAB. b. Find mBCD. SOLUTION a. mDAB + mABD = 90° mDAB + 35° = 90° mDAB + 35° – 35° = 90° – 35° mDAB = 55° b. mDAB + mBCD = 90° 55° + mBCD = 90° mBCD = 35° Corollary to the Triangle Sum Theorem Substitute 35° for mABD. Subtract 35° from each side. Simplify. Corollary to the Triangle Sum Theorem Substitute 55° for mDAB. Subtract 55° from each side. Your Turn: 1. Find mA. ANSWER 65° ANSWER 75° ANSWER 50° 2. Find mB. 3. Find mC. Example 6: GARDENING The flower bed shown is in the shape of a right triangle. Find if is 20. Corollary 4.1 Substitution Subtract 20 from each side. Answer: Your Turn: The piece of quilt fabric is in the shape of a right triangle. Find if is 62. Answer: Joke Time What's orange and sounds like a parrot? A carrot! What do you call cheese that doesn't belong to you? Nacho cheese. Why do farts smell? So the deaf can enjoy them too. Assignment IXL W6 IXL W11 IXL W 12 IXL W 15

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