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Transcript

Angles of Triangles Section 4.2 Objectives Find angle measures in triangles. Key Vocabulary Corollary Exterior angles Interior angles Theorems 4.1 Triangle Sum Theorem Corollary to the Triangle Sum Theorem 4.2 Exterior Angle Theorem 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 Each vertex has a pair of congruent exterior angles; however it is common to show only one exterior angle at each vertex. 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. Triangle Sum Theorem If you tear off two corners of a triangle and place them next to the third corner, the three angles seem to form a straight line. You can also show this in a drawing. Triangle Sum Theorem Draw a triangle and extend one side. Then draw a line parallel to the extended side, as shown. Two sides of the triangle are transversals to the parallel lines. The three angles in the triangle can be arranged to form a straight line or 180°. 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 –117 –117 p = 63 Triangle Sum Theorem Subtract 117 from both sides. Example 2b B. Find m in the obtuse triangle. 62 23° + 62° + m° = 180° Triangle Sum Theorem 23 85 + m = 180 –85 –85 m = 95 Subtract 85 from both sides. m Your Turn: A. Find a in the acute triangle. 88° + 38° + a° = 180° 126 + a = 180 –126 –126 a = 54 Triangle Sum Theorem 38° Subtract 126 from both sides. a° 88° Your Turn: B. Find c in the obtuse triangle. 24° + 38° + c° = 180° 62 + c = 180 –62 –62 c = 118 Triangle Sum Theorem. 38° 24° Subtract 62 from both sides. c° 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° 20x = 180 20 20 x=9 The angle labeled 3x° measures 3(9°) = 27°, the angle labeled 7x° measures 7(9°) = 63°, and the angle labeled 10x° measures 10(9°) = 90°. Triangle Sum Theorem Simplify. Divide both sides by 20. 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: Corollaries Definition: A corollary is a theorem with a proof that follows as a direct result of another theorem. As a theorem, a corollary can be used as a reason in a proof. 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: Exterior Angles and Triangles An exterior angle is formed by one side of a triangle and the extension of another side (i.e. 1 ). 2 1 4 3 The interior angles of the triangle not adjacent to a given exterior angle are called the remote interior angles (i.e. 2 and 3). Investigating Exterior Angles of a Triangles You can put the two torn angles together to exactly cover one of the exterior angles B A B C A Theorem 4.2 – Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. m 1 = m 2 + m 3 2 1 4 3 Example 7 Given mA = 58° and mC = 72°, find m1. SOLUTION m1 = mA + mC ANSWER Exterior Angle Theorem = 58° + 72° Substitute 58° for mA and 72° for mC. = 130° Simplify. 1 has a measure of 130°. Your Turn: 1. Find m2. ANSWER 120° ANSWER 155° ANSWER 113° 2. Find m3. 3. Find m4. Example 8: Find the measure of each numbered angle in the figure. Exterior Angle Theorem Simplify. If 2 s form a linear pair, they are supplementary. Substitution Subtract 70 from each side. Example 8: m∠1=70 m∠2=110 Exterior Angle Theorem Substitution Subtract 64 from each side. If 2 s form a linear pair, they are supplementary. Substitution Simplify. Subtract 78 from each side. m∠1=70 m∠2=110 m∠3=46 m∠4=102 Example 8: Angle Sum Theorem Substitution Simplify. Subtract 143 from each side. Answer: Your Turn: Find the measure of each numbered angle in the figure. 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 Pg. 182-184: #1 – 13 odd, 19 – 29 odd