The document discusses the Hinge Theorem and its converse for comparing sides and angles of triangles. It provides examples of applying the Hinge Theorem and its converse to determine if one side or angle is greater than the other. It also gives an example problem of proving that one side is less than the other using the Hinge Theorem and properties of alternate interior angles for parallel lines cut by a transversal. The document concludes with assigning practice problems related to applying the Hinge Theorem and its converse.

1. Section 5-6
Inequalities in Two Triangles
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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?
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3. Hinge Theorem
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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.
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5. Converse of the Hinge Theorem
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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.
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7. Example 1
Compare the given measures.
a. AD and BD
m∠ACD = 70°, m∠BCD = 68°
b. m∠ABD, m∠CDB
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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
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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
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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.
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12. Example 3
Find the range of possible values for a.
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13. Example 3
Find the range of possible values for a.
9a +15<141
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14. Example 3
Find the range of possible values for a.
9a +15<141
−15 −15
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15. Example 3
Find the range of possible values for a.
9a +15<141
−15 −15
9a <126
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16. Example 3
Find the range of possible values for a.
9a +15<141
−15 −15
9a <126
9 9
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17. Example 3
Find the range of possible values for a.
9a +15<141
−15 −15
9a <126
9 9
a <14
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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
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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
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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
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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
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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
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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
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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°.
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25. Example 4
Prove the following.
Given: JK = HL; JH || KL;
m∠JKH + m∠HKL < m∠JHK + m∠KHL
Prove: JH < KL
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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
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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
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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
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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
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30. Example 4
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31. Example 4
3. ∠HKL ≅ ∠JHK
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32. Example 4
3. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm
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33. Example 4
3. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm
4. m∠HKL = m∠JHK
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34. Example 4
3. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm
4. m∠HKL = m∠JHK 4. Definition of congruent
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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
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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
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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
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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
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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
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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
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41. Problem Set
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42. Problem Set
p. 371 #1-29 odd, 39, 47, 51
"Make visible what, without you, might perhaps never have been
seen." - Robert Bresson
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