Triangle Sum Theorem

triangle sum theorem

1. Angles of Triangles Section 4.2

2. Objectives Find angle measures in triangles.

3. Key Vocabulary Corollary Exterior angles Interior angles

4. Theorems 4.1 Triangle Sum Theorem Corollary to the Triangle Sum Theorem 4.2 Exterior Angle Theorem

5. 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 Interior Angle Exterior Angle Exterior Angle

6. Triangle Interior and Exterior Angles 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 and Exterior Angles )) ))) ( A B C ( )) (( D E F Interior Angles Exterior Angles (formed by extending the sides)

8. Triangle Sum Theorem The Triangle Angle-Sum Theorem gives the relationship among the interior angle measures of any triangle.

9. 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.

10. Draw a triangle and 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 °. m X + m Y + mZ = 180 ° X Y Z

12. Triangle Sum Theorem

13. Given m  A = 43 ° and m  B = 85 ° , find m  C . ANSWER  C has a measure of 52 ° . CHECK Check your solution by substituting 52 ° for m  C . 43 ° + 85 ° + 52 ° = 180 ° SOLUTION m  A + m  B + m  C = 180 ° Triangle Sum Theorem 43 ° + 85 ° + m  C = 180 ° Substitute 43 ° for m  A and 85 ° for m  B . 128 ° + m  C = 180 ° Simplify. m  C = 52 ° Simplify. 128 ° + m  C – 128 ° = 180 ° – 128 ° Subtract 128 ° from each side. Example 1

14. A. Find p in 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 m in 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 a in 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 c in 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. 2 x ° + 3 x ° + 5 x° = 180° 10 x = 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 2 x ° measures 2(18°) = 36°, the angle labeled 3 x° measures 3(18°) = 54°, and the angle labeled 5 x° measures 5(18°) = 90°. Example 3

19. 3 x ° + 7 x ° + 10 x° = 180° 20 x = 180 x = 9 20 20 Find the angle measures in the scalene triangle. Triangle Sum Theorem Simplify. Divide both sides by 20. 3 x ° 7 x ° 10 x° The angle labeled 3 x ° measures 3(9°) = 27°, the angle labeled 7 x ° measures 7(9°) = 63°, and the angle labeled 10 x ° measures 10(9°) = 90°. Your Turn:

20. 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:

21. Answer: Angle Sum Theorem Simplify. Subtract 142 from each side. Example 4:

22. Find the missing angle measures. Answer: Your Turn:

23. 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.

24. Triangle Angle-Sum Corollaries Corollary 4.1 – The acute  s of a right ∆ are complementary. Example: m ∠ x + m ∠ y = 90 ˚ x ° y °

25. m  DAB + 35 ° = 90 ° Substitute 35 ° for m  ABD . m  DAB = 55 ° Simplify. m  DAB + 35 ° – 35 ° = 90 ° – 35 ° Subtract 35 ° from each side. ∆ ABC and ∆ ABD are right triangles. Suppose m  ABD = 35° . Find m  DAB . a. b. Find m  BCD . 55 ° + m  BCD = 90 ° Substitute 55 ° for m  DAB . m  BCD = 35 ° Subtract 55 ° from each side. SOLUTION Corollary to the Triangle Sum Theorem m  DAB + m  ABD = 90 ° a. Corollary to the Triangle Sum Theorem m  DAB + m  BCD = 90 ° b. Example 5

26. ANSWER 65 ° ANSWER 75 ° ANSWER 50 ° Find m  A . 1. Find m  B . 2. Find m  C . 3. Your Turn:

27. Corollary 4.1 Substitution Subtract 20 from 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 of quilt fabric is in the shape of a right triangle. Find if is 6 2 . Your Turn:

29. Exterior Angles and Triangles 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 3 4

30. Investigating Exterior Angles of 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 3 4

32. ANSWER  1 has a measure of 130 ° . SOLUTION m 1 = m  A + m  C Exterior Angle Theorem Given m  A = 58 ° and m  C = 72 ° , find m  1 . Substitute 58 ° for m  A and 72 ° for m  C . = 58 ° + 72 ° Simplify. = 130 ° Example 7

33. ANSWER 120 ° ANSWER 155 ° ANSWER 113 ° Find m  2 . 1. Find m  3 . 2. Find m  4 . 3. Your Turn:

34. Find the measure of 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 Subtract 64 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 from each side. Angle Sum Theorem Substitution Simplify. Answer: Example 8: m ∠ 1=70 m ∠ 2=110 m ∠ 3=46 m ∠ 4=102

37. Find the measure of each numbered angle in the figure. Answer: Your Turn:

38. 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.

39. Assignment Pg. 182-184: #1 – 13 odd, 19 – 29 odd