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# Geometry Section 5-6 1112

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Inequalities in Two Triangles

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### Geometry Section 5-6 1112

1. 1. Section 5-6 Inequalities in Two Triangles Tuesday, April 15, 14
2. 2. Essential Questions How do you apply the Hinge Theorem or its converse to make comparisons in two triangles? How do you prove triangle relationships using the Hinge Theorem or its converse? Tuesday, April 15, 14
3. 3. Hinge Theorem Tuesday, April 15, 14
4. 4. Hinge Theorem If two sides of a triangle are congruent to two sides of another triangle, and the included angle of the ﬁrst is larger than the included angle of the second triangle, then the third side of the ﬁrst triangle is longer than the third side of the second triangle. Tuesday, April 15, 14
5. 5. Converse of the Hinge Theorem Tuesday, April 15, 14
6. 6. Converse of the Hinge Theorem If two sides of a triangle are congruent to two sides of another triangle, and the third side of the ﬁrst triangle is longer than the third side of the second triangle, then the included angle measure of the ﬁrst is larger than the included angle measure of the second triangle. Tuesday, April 15, 14
7. 7. Example 1 Compare the given measures. a. AD and BD m∠ACD = 70°, m∠BCD = 68° b. m∠ABD, m∠CDB Tuesday, April 15, 14
8. 8. Example 1 Compare the given measures. a. AD and BD m∠ACD = 70°, m∠BCD = 68° AD > BD by the Hinge Theorem b. m∠ABD, m∠CDB Tuesday, April 15, 14
9. 9. Example 1 Compare the given measures. a. AD and BD m∠ACD = 70°, m∠BCD = 68° AD > BD by the Hinge Theorem b. m∠ABD, m∠CDB By the Converse of the Hinge Theorem, m∠ABD > m∠CDB Tuesday, April 15, 14
10. 10. Example 2 Doctors use a straight-leg-raising test to determine the amount of pain felt in a person’s back. The patient lies ﬂat on the examining table, and the doctor raises each leg until the patient experiences pain in the back area. Matt Mitarnowski can tolerate the doctor raising his right leg 35° and his left leg 65° from the table. Which leg can Matt raise higher above the table? How do you know? Tuesday, April 15, 14
11. 11. Example 2 Doctors use a straight-leg-raising test to determine the amount of pain felt in a person’s back. The patient lies ﬂat on the examining table, and the doctor raises each leg until the patient experiences pain in the back area. Matt Mitarnowski can tolerate the doctor raising his right leg 35° and his left leg 65° from the table. Which leg can Matt raise higher above the table? How do you know? As the angle between the table and leg is greater, Matt can lift his left leg higher. Tuesday, April 15, 14
12. 12. Example 3 Find the range of possible values for a. Tuesday, April 15, 14
13. 13. Example 3 Find the range of possible values for a. 9a +15<141 Tuesday, April 15, 14
14. 14. Example 3 Find the range of possible values for a. 9a +15<141 −15 −15 Tuesday, April 15, 14
15. 15. Example 3 Find the range of possible values for a. 9a +15<141 −15 −15 9a <126 Tuesday, April 15, 14
16. 16. Example 3 Find the range of possible values for a. 9a +15<141 −15 −15 9a <126 9 9 Tuesday, April 15, 14
17. 17. Example 3 Find the range of possible values for a. 9a +15<141 −15 −15 9a <126 9 9 a <14 Tuesday, April 15, 14
18. 18. Example 3 Find the range of possible values for a. 9a +15<141 −15 −15 9a <126 9 9 a <14 9a +15>0 Tuesday, April 15, 14
19. 19. Example 3 Find the range of possible values for a. 9a +15<141 −15 −15 9a <126 9 9 a <14 9a +15>0 −15 −15 Tuesday, April 15, 14
20. 20. Example 3 Find the range of possible values for a. 9a +15<141 −15 −15 9a <126 9 9 a <14 9a +15>0 −15 −15 9a > −15 Tuesday, April 15, 14
21. 21. Example 3 Find the range of possible values for a. 9a +15<141 −15 −15 9a <126 9 9 a <14 9a +15>0 −15 −15 9a > −15 9 9 Tuesday, April 15, 14
22. 22. Example 3 Find the range of possible values for a. 9a +15<141 −15 −15 9a <126 9 9 a <14 9a +15>0 −15 −15 9a > −15 9 9 a > − 5 3 Tuesday, April 15, 14
23. 23. Example 3 Find the range of possible values for a. 9a +15<141 −15 −15 9a <126 9 9 a <14 9a +15>0 −15 −15 9a > −15 9 9 a > − 5 3 − 5 3 <a <14 Tuesday, April 15, 14
24. 24. Example 3 Find the range of possible values for a. 9a +15<141 −15 −15 9a <126 9 9 a <14 9a +15>0 −15 −15 9a > −15 9 9 a > − 5 3 − 5 3 <a <14 If we are looking at the smaller angle, it must be larger than 0°. If we are looking at the larger angle, it must be less than 180°. Tuesday, April 15, 14
25. 25. Example 4 Prove the following. Given: JK = HL; JH || KL; m∠JKH + m∠HKL < m∠JHK + m∠KHL Prove: JH < KL Tuesday, April 15, 14
26. 26. Example 4 Prove the following. 1. JK = HL; JH || KL; m∠JKH + m∠HKL < m∠JHK + m∠KHL Given: JK = HL; JH || KL; m∠JKH + m∠HKL < m∠JHK + m∠KHL Prove: JH < KL Tuesday, April 15, 14
27. 27. Example 4 Prove the following. 1. Given 1. JK = HL; JH || KL; m∠JKH + m∠HKL < m∠JHK + m∠KHL Given: JK = HL; JH || KL; m∠JKH + m∠HKL < m∠JHK + m∠KHL Prove: JH < KL Tuesday, April 15, 14
28. 28. Example 4 Prove the following. 1. Given 2. HK ≅ HK 1. JK = HL; JH || KL; m∠JKH + m∠HKL < m∠JHK + m∠KHL Given: JK = HL; JH || KL; m∠JKH + m∠HKL < m∠JHK + m∠KHL Prove: JH < KL Tuesday, April 15, 14
29. 29. Example 4 Prove the following. 1. Given 2. Reﬂexive2. HK ≅ HK 1. JK = HL; JH || KL; m∠JKH + m∠HKL < m∠JHK + m∠KHL Given: JK = HL; JH || KL; m∠JKH + m∠HKL < m∠JHK + m∠KHL Prove: JH < KL Tuesday, April 15, 14
30. 30. Example 4 Tuesday, April 15, 14
31. 31. Example 4 3. ∠HKL ≅ ∠JHK Tuesday, April 15, 14
32. 32. Example 4 3. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm Tuesday, April 15, 14
33. 33. Example 4 3. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm 4. m∠HKL = m∠JHK Tuesday, April 15, 14
34. 34. Example 4 3. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm 4. m∠HKL = m∠JHK 4. Deﬁnition of congruent Tuesday, April 15, 14
35. 35. Example 4 3. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm 5. m∠JKH + m∠JHK < m∠JHK + m∠KHL 4. m∠HKL = m∠JHK 4. Deﬁnition of congruent Tuesday, April 15, 14
36. 36. Example 4 3. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm 5. m∠JKH + m∠JHK < m∠JHK + m∠KHL 5. Substitution 4. m∠HKL = m∠JHK 4. Deﬁnition of congruent Tuesday, April 15, 14
37. 37. Example 4 3. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm 5. m∠JKH + m∠JHK < m∠JHK + m∠KHL 5. Substitution 6. m∠JKH < m∠KHL 4. m∠HKL = m∠JHK 4. Deﬁnition of congruent Tuesday, April 15, 14
38. 38. Example 4 3. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm 5. m∠JKH + m∠JHK < m∠JHK + m∠KHL 5. Substitution 6. m∠JKH < m∠KHL 6. Subtraction 4. m∠HKL = m∠JHK 4. Deﬁnition of congruent Tuesday, April 15, 14
39. 39. Example 4 3. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm 5. m∠JKH + m∠JHK < m∠JHK + m∠KHL 5. Substitution 7. JH < KL 6. m∠JKH < m∠KHL 6. Subtraction 4. m∠HKL = m∠JHK 4. Deﬁnition of congruent Tuesday, April 15, 14
40. 40. Example 4 3. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm 5. m∠JKH + m∠JHK < m∠JHK + m∠KHL 5. Substitution 7. JH < KL 7. Hinge Theorem 6. m∠JKH < m∠KHL 6. Subtraction 4. m∠HKL = m∠JHK 4. Deﬁnition of congruent Tuesday, April 15, 14
41. 41. Problem Set Tuesday, April 15, 14
42. 42. Problem Set p. 371 #1-29 odd, 39, 47, 51 "Make visible what, without you, might perhaps never have been seen." - Robert Bresson Tuesday, April 15, 14