Geometry Section 5-6 1112
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Transcript of Geometry Section 5-6 1112
Section 5-6Inequalities 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 1Compare the given measures.
a. AD and BD
m∠ACD = 70°, m∠BCD = 68°
b. m∠ABD, m∠CDB
Tuesday, April 15, 14
Example 1Compare the given measures.
a. AD and BD
m∠ACD = 70°, m∠BCD = 68°
AD > BD by theHinge Theorem
b. m∠ABD, m∠CDB
Tuesday, April 15, 14
Example 1Compare the given measures.
a. AD and BD
m∠ACD = 70°, m∠BCD = 68°
AD > BD by theHinge Theorem
b. m∠ABD, m∠CDB
By the Converse of the Hinge Theorem,
m∠ABD > m∠CDBTuesday, April 15, 14
Example 2Doctors 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 2Doctors 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 3Find the range of possible values for a.
Tuesday, April 15, 14
Example 3Find the range of possible values for a.
9a +15<141
Tuesday, April 15, 14
Example 3Find the range of possible values for a.
9a +15<141 −15 −15
Tuesday, April 15, 14
Example 3Find the range of possible values for a.
9a +15<141 −15 −15
9a <126
Tuesday, April 15, 14
Example 3Find the range of possible values for a.
9a +15<141 −15 −15
9a <126 9 9
Tuesday, April 15, 14
Example 3Find the range of possible values for a.
9a +15<141 −15 −15
9a <126 9 9 a <14
Tuesday, April 15, 14
Example 3Find 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 3Find 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 3Find 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 3Find 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 3Find 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 >−
53
Tuesday, April 15, 14
Example 3Find 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 >−
53
−
53<a <14
Tuesday, April 15, 14
Example 3Find 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 >−
53
−
53<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 4Prove the following.
Given: JK = HL; JH || KL;m∠JKH + m∠HKL < m∠JHK + m∠KHL
Prove: JH < KL
Tuesday, April 15, 14
Example 4Prove 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 4Prove the following.
1. Given1. 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 4Prove 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 4Prove 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 43. ∠HKL ≅ ∠JHK
Tuesday, April 15, 14
Example 43. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm
Tuesday, April 15, 14
Example 43. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm
4. m∠HKL = m∠JHK
Tuesday, April 15, 14
Example 43. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm
4. m∠HKL = m∠JHK 4. Definition of congruent
Tuesday, April 15, 14
Example 43. ∠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 43. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm
5. m∠JKH + m∠JHK < m∠JHK + m∠KHL5. Substitution
4. m∠HKL = m∠JHK 4. Definition of congruent
Tuesday, April 15, 14
Example 43. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm
5. m∠JKH + m∠JHK < m∠JHK + m∠KHL5. Substitution
6. m∠JKH < m∠KHL
4. m∠HKL = m∠JHK 4. Definition of congruent
Tuesday, April 15, 14
Example 43. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm
5. m∠JKH + m∠JHK < m∠JHK + m∠KHL5. Substitution
6. m∠JKH < m∠KHL 6. Subtraction
4. m∠HKL = m∠JHK 4. Definition of congruent
Tuesday, April 15, 14
Example 43. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm
5. m∠JKH + m∠JHK < m∠JHK + m∠KHL5. 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 43. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm
5. m∠JKH + m∠JHK < m∠JHK + m∠KHL5. 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