Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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