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?

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.

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.

Example 1Compare the given measures.

a. AD and BD

m∠ACD = 70°, m∠BCD = 68°

b. m∠ABD, m∠CDB

a. AD and BD

m∠ACD = 70°, m∠BCD = 68°

AD > BD by theHinge Theorem

b. m∠ABD, m∠CDB

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?

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.

Example 3Find the range of possible values for a.

9a +15<141

9a +15<141 −15 −15

9a <126

9a +15<141 −15 −15

9a <126 9 9

9a +15<141 −15 −15

9a <126 9 9 a <14

9a +15<141 −15 −15

9a <126 9 9 a <14

9a +15> 0

9a +15<141 −15 −15

9a <126 9 9 a <14

9a +15> 0 −15 −15

9a +15<141 −15 −15

9a <126 9 9 a <14

9a +15> 0 −15 −15 9a >−15

9a +15<141 −15 −15

9a <126 9 9 a <14

9a +15> 0 −15 −15 9a >−15 9 9

9a +15<141 −15 −15

9a <126 9 9 a <14

9a +15> 0 −15 −15 9a >−15 9 9

a >−

9a +15<141 −15 −15

9a <126 9 9 a <14

9a +15> 0 −15 −15 9a >−15 9 9

a >−

53<a <14

9a +15<141 −15 −15

9a <126 9 9 a <14

9a +15> 0 −15 −15 9a >−15 9 9

a >−

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°.

Example 4Prove the following.

Given: JK = HL; JH || KL;m∠JKH + m∠HKL < m∠JHK + m∠KHL

Prove: JH < KL

1. JK = HL; JH || KL;m∠JKH + m∠HKL < m∠JHK + m∠KHL

Prove: JH < KL

1. Given1. JK = HL; JH || KL;

m∠JKH + m∠HKL < m∠JHK + m∠KHL

Prove: JH < KL

1. Given

2. HK ≅ HK

Prove: JH < KL

1. Given

2. Reflexive2. HK ≅ HK

Prove: JH < KL

Example 4

Example 43. ∠HKL ≅ ∠JHK

Example 43. ∠HKL ≅ ∠JHK 3. Alt. Int. Angles Thm

4. m∠HKL = m∠JHK

4. m∠HKL = m∠JHK 4. Definition of congruent

5. m∠JKH + m∠JHK < m∠JHK + m∠KHL

5. m∠JKH + m∠JHK < m∠JHK + m∠KHL5. Substitution

6. m∠JKH < m∠KHL

6. m∠JKH < m∠KHL 6. Subtraction

7. JH < KL

7. JH < KL 7. Hinge Theorem

Problem Set

p. 371 #1-29 odd, 39, 47, 51

"Make visible what, without you, might perhaps never have been seen." - Robert Bresson

Geometry Section 5-6 1112

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