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Section 5-6
Inequalities in Two Triangles
Tuesday, April 15, 14
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
Hinge Theorem
Tuesday, April 15, 14
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
Converse of the Hinge Theorem
Tuesday, April 15, 14
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
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
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
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
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
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
Example 3
Find the range of possible values for a.
Tuesday, April 15, 14
Example 3
Find the range of possible values for a.
9a +15<141
Tuesday, April 15, 14
Example 3
Find the range of possible values for a.
9a +15<141
−15 −15
Tuesday, April 15, 14
Example 3
Find the range of possible values for a.
9a +15<141
−15 −15
9a <126
Tuesday, April 15, 14
Example 3
Find the range of possible values for a.
9a +15<141
−15 −15
9a <126
9 9
Tuesday, April 15, 14
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
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
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
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
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
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
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
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
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
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
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
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
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
Example 4
Tuesday, April 15, 14
Example 4
3. ∠HKL ≅ ∠JHK
Tuesday, April 15, 14
Example 4
3. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm
Tuesday, April 15, 14
Example 4
3. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm
4. m∠HKL = m∠JHK
Tuesday, April 15, 14
Example 4
3. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm
4. m∠HKL = m∠JHK 4. Definition of congruent
Tuesday, April 15, 14
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
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
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
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
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
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
Problem Set
Tuesday, April 15, 14
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

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Comparing Sides and Angles in Triangles Using Hinge Theorem

  • 1. Section 5-6 Inequalities in Two Triangles Tuesday, April 15, 14
  • 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
  • 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. Converse of the Hinge Theorem Tuesday, April 15, 14
  • 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. 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. 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. 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. 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. 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. Example 3 Find the range of possible values for a. Tuesday, April 15, 14
  • 13. Example 3 Find the range of possible values for a. 9a +15<141 Tuesday, April 15, 14
  • 14. Example 3 Find the range of possible values for a. 9a +15<141 −15 −15 Tuesday, April 15, 14
  • 15. Example 3 Find the range of possible values for a. 9a +15<141 −15 −15 9a <126 Tuesday, April 15, 14
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
  • 31. Example 4 3. ∠HKL ≅ ∠JHK Tuesday, April 15, 14
  • 32. Example 4 3. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm Tuesday, April 15, 14
  • 33. Example 4 3. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm 4. m∠HKL = m∠JHK Tuesday, April 15, 14
  • 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. 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. 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. 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. 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. 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. 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
  • 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