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5.6Inequalities in Two Triangles Theorem 5.14: Hinge Theorem If two sides of one triangle are...

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5.6 Inequalities in Two Triangles Theorem 5.14: Hinge Theorem _____ WX ST If two sides of one 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, then the third side of the first is _________ than the third side of the second. longer V X W 88 R S T 35
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Page 1: 5.6Inequalities in Two Triangles Theorem 5.14: Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included.

5.6 Inequalities in Two Triangles

Theorem 5.14: Hinge Theorem

_____WX ST

If two sides of one 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, then the third side of the first is _________ than the third side of the second.

longer

V

X

W

88R

S

T

35

Page 2: 5.6Inequalities in Two Triangles Theorem 5.14: Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included.

5.6 Inequalities in Two Triangles

Theorem 5.15: Converse of the Hinge Theorem

_____Cm Fm

If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is _________ than the included angle of the second.

larger

A

C B

12

D

EF

9

Page 3: 5.6Inequalities in Two Triangles Theorem 5.14: Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included.

5.6 Inequalities in Two Triangles

Example 1 Use the Hinge Theorem and its converseUse the Hinge Theorem and its converse

Complete the statement with <, >, or =. Explain.

EF ____ AC a. CBD ____ ADB b. mmB

C

A61

E

D

F

62

A

D

B

C

34

33

a. You are given thatSolution

.____BC and ____AB Because 61o < ____, by the Hinge Theorem, AC < ____.

ED DFo62 EF

? ?

Page 4: 5.6Inequalities in Two Triangles Theorem 5.14: Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included.

5.6 Inequalities in Two Triangles

Example 1 Use the Hinge Theorem and its converseUse the Hinge Theorem and its converse

Complete the statement with <, >, or =. Explain.

EF ____ AC a. CBD ____ ADB b. mmB

C

A61

E

D

F

62

A

D

B

C

34

33

b. You are given thatSolution

____AD by the Reflexive Property. Because 34 > 33, ____ > ____. So, by the Converse of the Hinge Theorem, ________ > ________.

BC BDAB CD

and you know that ____BD

ADBm CBDm

? ?

Page 5: 5.6Inequalities in Two Triangles Theorem 5.14: Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included.

5.6 Inequalities in Two Triangles

Example 2 Solve a multi-step problemSolve a multi-step problem

TravelTravel Car A leaves a mall, heads due north for 5 mi and then turns due west for 3 mi. Car B leaves the same mall, heads due south for 5 mi and then turns 80o towards east for 3 mi. Which car is farther from the mall?Draw a diagram.

Mall

o80

o90

mi 5

mi 3

mi 5

mi 3

The distance driven and the distance back to the mall form two triangles, with _________ 5 mile sides and __________ 3 mile sides.

congruentcongruent

Add the third side to the triangle.Use linear pairs to find the included angles of _____ and _____.o90

o90

o100o100

Because 100o > 90o, Car ___ is farther from the mall than Car A by the _______________.

BHinge Theorem

B

A

Page 6: 5.6Inequalities in Two Triangles Theorem 5.14: Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included.

5.6 Inequalities in Two Triangles

Example 3 Write an indirect proofWrite an indirect proof

Write an indirect proof to show that an odd number is not Write an indirect proof to show that an odd number is not divisible by 6divisible by 6.

Given x is an odd number.Prove x is not divisible by 6.

Assume temporarily that ______________________.

This means that ____ = n for some whole number n.

x is divisible by 6

6

xStep 1Step 1Solution

So, multiplying both sides by 6 gives ___ = ___. x 6n

Page 7: 5.6Inequalities in Two Triangles Theorem 5.14: Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included.

5.6 Inequalities in Two Triangles

Example 3 Write an indirect proofWrite an indirect proof

Write an indirect proof to show that an odd number is not Write an indirect proof to show that an odd number is not divisible by 6divisible by 6.

Given x is an odd number.Prove x is not divisible by 6.

SolutionIf x is odd, then by definition, x cannot be divided evenly

by ___. However, __ = ___ so ___ = ____ = ___. 2 2

xStep 2Step 2

x 6n 2

6n3n

We know that __ is a whole number because n is a whole number, so x can be divided evenly by __.

3n2

This contradicts the given statement that _________. x is odd

Therefore, the assumption that x is divisible by 6 is _____,Step 3Step 3 false

which proves that _______________________.x is not divisible by 6

Page 8: 5.6Inequalities in Two Triangles Theorem 5.14: Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included.

5.6 Inequalities in Two Triangles Checkpoint. Complete the following exercises. Checkpoint. Complete the following exercises.

1. CDB,ADB If mmwhich is longer, AB or CB?

A

B

DC91 89

longer is AB

Page 9: 5.6Inequalities in Two Triangles Theorem 5.14: Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included.

5.6 Inequalities in Two Triangles Checkpoint. Complete the following exercises. Checkpoint. Complete the following exercises.

In example 2, car C leaves the mall, and goes 5 miles due west then turns 85o towards south for 3 mi. Write the cars in order from the car closest to the mall to the car farthest from the mall.

Mall

o80

o90

mi 5

mi 3

mi 5

mi 3

o90

o100 B

A

o85mi 3

mi 5o95

C

Car A is closest

Car C is next closest

Car B is farthest

Page 10: 5.6Inequalities in Two Triangles Theorem 5.14: Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included.

5.6 Inequalities in Two Triangles Checkpoint. Complete the following exercises. Checkpoint. Complete the following exercises.

3. Suppose you want to prove the statement “If x + y = 5 and y = 2, then x = 3.” What temporary assumption could you make to prove the conclusion indirectly?

You can temporarily assume that

3x

Page 11: 5.6Inequalities in Two Triangles Theorem 5.14: Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included.

5.6 Inequalities in Two Triangles

Pg. 305, 5.6 #1-15


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