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Triangle Sum Theorem
- 1.
- 2. Objectives Find angle measuresin triangles.
- 3.
- 4. Theorems 4.1 Triangle SumTheorem Corollary to the Triangle Sum Theorem 4.2 Exterior Angle Theorem
- 5. Measures of Anglesof 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 Interior Angle Exterior Angle Exterior Angle
- 6. Triangle Interior andExterior Angles A B C Smiley faces are interior angles and hearts represent the exterior angles Each vertex has a pair of congruent exterior angles; however it is common to show only one exterior angle at each vertex.
- 7. Triangle Interior andExterior Angles ))) A B C ( D E F Interior Angles Exterior Angles (formed by extending the sides)
- 8. Triangle Sum Theorem TheTriangle Angle-Sum Theorem gives the relationship among the interior angle measures of any triangle.
- 9. Triangle Sum Theorem Ifyou 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.
- 10. Draw a triangleand extend one side. Then draw a line parallel to the extended side, as shown. The three angles in the triangle can be arranged to form a straight line or 180°. Two sides of the triangle are transversals to the parallel lines. Triangle Sum Theorem
- 11. Theorem 4.1 –Triangle Sum Theorem The sum of the measures of the angles of a triangle is 180°. mX + mY + mZ = 180° X Y Z
- 12.
- 13. Given mA =43° and mB = 85°, find mC. ANSWER C has a measure of 52°. CHECK Check your solution by substituting 52° for mC. 43° + 85° + 52° = 180° SOLUTION mA + mB + mC = 180° Triangle Sum Theorem 43° + 85° + mC = 180° Substitute 43° for mA and 85° for mB. 128° + mC = 180° Simplify. mC = 52° Simplify. 128° + mC – 128° = 180° – 128° Subtract 128° from each side. Example 1
- 14. A. Find pin the acute triangle. 73° + 44° + p° = 180° 117 + p = 180 p = 63 –117 –117 Triangle Sum Theorem Subtract 117 from both sides. Example 2a
- 15. B. Find min the obtuse triangle. 23° + 62° + m° = 180° 85 + m = 180 m = 95 –85 –85 Triangle Sum Theorem Subtract 85 from both sides. 23 62 m Example 2b
- 16. A. Find ain the acute triangle. 88° + 38° + a° = 180° 126 + a = 180 a = 54 –126 –126 88° 38° a° Triangle Sum Theorem Subtract 126 from both sides. Your Turn:
- 17. B. Find cin the obtuse triangle. 24° + 38° + c° = 180° 62 + c = 180 c = 118 –62 –62 c° 24° 38° Triangle Sum Theorem. Subtract 62 from both sides. Your Turn:
- 18. 2x° + 3x°+ 5x° = 180° 10x = 180 x = 18 10 10 Find the angle measures in the scalene triangle. Triangle Sum Theorem Simplify. Divide both sides by 10. 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°. Example 3
- 19. 3x° + 7x°+ 10x° = 180° 20x = 180 x = 9 20 20 Find the angle measures in the scalene triangle. Triangle Sum Theorem Simplify. Divide both sides by 20. 3x° 7x° 10x°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°. Your Turn:
- 20. Find the missingangle 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:
- 21. Answer: Angle Sum Theorem Simplify. Subtract142 from each side. Example 4:
- 22. Find the missingangle measures. Answer: Your Turn:
- 23. Corollaries Definition: A corollaryis 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.
- 24. Triangle Angle-Sum Corollaries Corollary4.1 – The acute s of a right ∆ are complementary. Example: m∠x + m∠y = 90˚ x° y°
- 25. mDAB + 35°= 90° Substitute 35° for mABD. mDAB = 55° Simplify. mDAB + 35° – 35° = 90° – 35° Subtract 35° from each side. ∆ABC and ∆ABD are right triangles. Suppose mABD = 35°. Find mDAB.a. b.Find mBCD. 55° + mBCD = 90° Substitute 55° for mDAB. mBCD = 35° Subtract 55° from each side. SOLUTION Corollary to the Triangle Sum Theorem mDAB + mABD = 90°a. Corollary to the Triangle Sum Theorem mDAB + mBCD = 90°b. Example 5
- 26. ANSWER 65° ANSWER 75° ANSWER50° Find mA.1. Find mB.2. Find mC.3. Your Turn:
- 27. Corollary 4.1 Substitution Subtract 20from each side. Answer: GARDENING The flower bed shown is in the shape of a right triangle. Find if is 20. Example 6:
- 28. Answer: The piece ofquilt fabric is in the shape of a right triangle. Find if is 62. Your Turn:
- 29. Exterior Angles andTriangles An exterior angle is formed by one side of a triangle and the extension of another side (i.e. 1 ). The interior angles of the triangle not adjacent to a given exterior angle are called the remote interior angles (i.e. 2 and 3). 1 2 34
- 30. Investigating Exterior Anglesof a Triangles B A A B C You can put the two torn angles together to exactly cover one of the exterior angles
- 31. 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 1 2 34
- 32. ANSWER 1 hasa measure of 130°. SOLUTION m1 = mA + mC Exterior Angle Theorem Given mA = 58° and mC = 72°, find m1. Substitute 58° for mA and 72° for mC. = 58° + 72° Simplify.= 130° Example 7
- 33. ANSWER 120° ANSWER 155° ANSWER113° Find m2.1. Find m3.2. Find m4.3. Your Turn:
- 34. Find the measureof each numbered angle in the figure. Exterior Angle Theorem Simplify. Substitution Subtract 70 from each side. If 2 s form a linear pair, they are supplementary. Example 8:
- 35. Exterior Angle Theorem Subtract64 from each side. Substitution Subtract 78 from each side. If 2 s form a linear pair, they are supplementary. Substitution Simplify. Example 8: m∠1=70 m∠2=110
- 36. Subtract 143 fromeach side. Angle Sum Theorem Substitution Simplify. Answer: Example 8: m∠1=70 m∠2=110 m∠3=46 m∠4=102
- 37. Find the measureof each numbered angle in the figure. Answer: Your Turn:
- 38. Joke Time What's orangeand 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.
- 39. Assignment Pg. 182-184: #1– 13 odd, 19 – 29 odd