Chapter 5: Relationships in Triangles

1. Chapter 5: Relationships in Triangles

2. Lesson 5.1 Bisectors, Medians, and Altitudes

3. Perpendicular Bisector A E B D C BD = CD AD BC E is the circumcenter- AE = BE = CE

4. Median A E B D C BD = CD E is the centroid- ED = 1/3 AD AE = 2/3 AD 2 ED = AE

5. Angle Bisector A E F G B D C BAD = CAD G is the incenter- EG = FG

6. Altitude A B D C AD BC

7. C. Find the measure of EH.

8. A. Find QS. B. FindWYZ.

9. In the figure, A is the circumcenter of ΔLMN. Find x if mAPM = 7x + 13.

10. In the figure, point D is the incenter of ΔABC. What segment is congruent to DG?

11. In ΔXYZ, P is the centroid and YV = 12. Find YP and PV.

12. In ΔLNP, R is the centroid and LO = 30. Find LR and RO.

14. Lesson 5.2 Inequalities and Triangles

15. Foldable • Fold the paper into three sections (burrito fold) Then fold the top edge down about ½ an inch • Unfold the paper and in the top small rectangles label each column…

19. List the angles of ΔABC in order from smallest to largest.

20. List the sides of ΔABC in order from shortest to longest.

21. What is the relationship between the lengths of RS and ST? ___ ___

22. What is the relationship between the measures of A and B?

23. Lesson 5.4 The Triangle Inequality

24. Triangle Inequality Theorem • The sum of the lengths of any two sides of a triangle is greater than the length of the third side

25. Determine if the measures given could be the sides of a triangle. 16, 17, 19 16 + 17 = 33 yes, the sum of the two smallest sides is larger than the third side 6, 9, 15 6 + 9 = 15 no, the sum of the two smallest sides is equal to the other side so it cannot be a triangle Find the range for the measure of the third side given the measures of two sides. 7.5 and 12.1 12.1- 7.5 < x < 12.1 + 7.5 4.6 < x < 19.6 9 and 41 41-9 < x < 41 + 9 32 < x < 50 Triangle Inequality Theorem Problems

26. Determine whether it is possible to form a triangle with side lengths 5, 7, and 8.

27. Is it possible to form a triangle with the given side lengths of 6.8, 7.2, 5.1? If not, explain why not.

28. Find the range for the measure of the third side of a triangle if two sides measure 4 and 13.

29. In ΔPQR, PQ = 7.2 and QR = 5.2. Which measure cannot be PR? A 7 B 9 C 11 D 13

30. Lesson 5.3 Indirect Proof

31. Steps to Completing an Indirect Proof: • Assume that ______________ (the conclusion is false) • Then _______________ (show that the assumption leads to a contradiction) This contradicts the given information that ________________. • Therefore, __________________ (rewrite the conclusion) must be true.

32. B. State the assumption you would make to start an indirect proof for the statement 3x = 4y + 1.

33. Example Indirect Proof Given: 5x < 25 Prove: x < 5 1. Assume that x 5. 2. Then x= 9 And 5(9)= 45 45> 25 This contradicts the given info that 5x < 25 3. Therefore, x < 5 must be true.

34. Example Indirect Proof m Given: m is not parallel to n Prove: m 3 m 2 3 2 n 1. Assume that m 3 = m 2 2. Then, angles 2 and 3 are alternate interior angles When alternate interior angles are congruent then the lines that make them are parallel. This contradicts the given info that m is not parallel to n 3. Therefore, m 3 m 2 must be true.

35. Write an indirect proof to show that if –2x + 11 < 7, then x > 2. Given: –2x + 11 < 7 Prove: x > 2

36. Write an indirect proof. Given:ΔJKLwith side lengths 5, 7, and 8 as shown. Prove:mK < mL

37. Lesson 5.5 Inequalities Involving Two Triangles

38. On the other side of the foldable from Lesson 2 (3 column chart)

39. A. Compare the measures AD and BD.

40. B. Compare the measures ABD and BDC.

41. ALGEBRA Find the range of possible values for a.

42. Find the range of possible values of n.