5.5: Inequalities in Two Triangles

5.5: Inequalities in Two Triangles

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Explore and Reason

A woodworker uses a caliper to measure the widths of a bat to help him determine
the widths for a new bat. The woodworker places the open tips of the caliper on
the bat. The distance between the tips is a width of the bat.

Explore and Reason

Suppose a caliper opens to an angle of 25° for one width of the red bat and opens to an angle of 35° for the blue bat. What can you conclude about the widths of the bats?

Next, suppose you use a caliper to measure the width of a narrow part of a
bat and a wider part of the bat. What can you predict about the angle to which the caliper opens each time?

Explore and Reason

Multiple Choice

Which end of the bat will have a wider caliper measurement?

• the end you hit the ball with

• the end you hold

EXAMPLE 1 Investigate Side Lengths in Triangles

As a rider pedals a unicycle, how do the measure of ∠𝘼 and length b change?

What does this suggest about the change in the triangle?

Dropdown

As m∠𝘼 increases, length b​

Select answer

EXAMPLE 1 Investigate Side Lengths in Triangles

In which of these triangles is m∠J the greatest?

In which of these triangles is m∠J the least?

In which of these triangles is k the greatest?

In which of these triangles is k the least?

Reorder

Place the triangles in order of the m∠J from least to greatest.

1

2

3

• 1
• 2
• 3

Multiple Choice

In which of these triangles is m∠J the least?

• acute

• obtuse

• right

Reorder

Place the triangles in order of the length of side k from least to greatest.

1

2

3

• 1
• 2
• 3

THEOREM 5-12 Hinge Theorem

If two sides of one triangle are congruent to two sides of another triangle, and the
included angles are not congruent, then the longer third side is opposite the larger
included angle.

EXAMPLE 2 Apply the Hinge Theorem

The tension in the exercise band varies proportionally with the stretch distance.
The tension T is described by the function 𝙏(𝙭)=𝙠𝙭, where k is a constant that
depends on the elasticity of the band and x is the stretch distance.
Which position shown in the figures has a greater tension in the band?

EXAMPLE 2 Apply the Hinge Theorem

T(LK) is the tension when the angle is 50° and 𝘛(𝘔𝘒) is the tension when the angle is 80°. Since 𝘮∠𝘔𝘑𝘒 > 𝘮∠𝘓𝘑𝘒, apply the Hinge Theorem.

EXAMPLE 2 Apply the Hinge Theorem

T(LK) is the tension when the angle is 50° and 𝘛(𝘔𝘒) is the tension when the angle is 80°. Since 𝘮∠𝘔𝘑𝘒 > 𝘮∠𝘓𝘑𝘒, apply the Hinge Theorem.

​A larger angle corresponds to a larger distance from the man's hands to his feet. The larger distance corresponds to a higher tension. The tension is greater when the man pulls higher on the tension band.

EXAMPLE 2 Apply the Hinge Theorem

The man keeps his arms extended and the length of the tension band the same. If he wants to make the measure of ∠𝘓 smaller, how would 𝘑𝘒 change?

ADDITIONAL EXAMPLE 2

Write an inequality relating BC and EF.

Dropdown

BC ​

Select answer

EF

ADDITIONAL EXAMPLE 2

Write an inequality relating BC and EF.

The Hinge Theorem states that, when two triangles have two pairs of congruent
sides, and the angles between those sides are unequal, the pair of sides opposite
those angles are also unequal. The side opposite the smaller angle is shorter than
the side opposite the larger angle.

In this pair of triangles, AB ≅ DE, AC ≅ DF, and 45° < 60°.

So, by the Hinge Theorem, BC < EF.

Theorem 5-13: Converse of the Hinge Theorem

If two sides of one triangle are congruent to two sides of another triangle, and the
third sides are not congruent, then the larger included angle is opposite the longer
third side.

Ex 3: Prove the Converse of the Hinge Theorem

Use indirect reasoning to prove the Converse of the Hinge Theorem.

Ex 3: Prove the Converse of the Hinge Theorem

Use indirect reasoning to prove the Converse of the Hinge Theorem.

Assume that 𝘮∠𝘍𝘋𝘌 is not greater than 𝘮∠𝘝𝘛𝘜, that is, that 𝘮∠𝘍𝘋𝘌 = 𝘮∠𝘝𝘛𝘜, or 𝘮∠𝘍𝘋𝘌 < 𝘮∠𝘝𝘛𝘜.

Ex 3: Prove the Converse of the Hinge Theorem

Use indirect reasoning to prove the Converse of the Hinge Theorem.

Assume that 𝘮∠𝘍𝘋𝘌 is not greater than 𝘮∠𝘝𝘛𝘜, that is, that 𝘮∠𝘍𝘋𝘌 = 𝘮∠𝘝𝘛𝘜, or 𝘮∠𝘍𝘋𝘌 < 𝘮∠𝘝𝘛𝘜.

Assuming that 𝘮∠𝘍𝘋𝘌=𝘮∠𝘝𝘛𝘜, ∠𝘍𝘋𝘌≅∠𝘝𝘛𝘜. Applying SAS, 𝘋𝘌𝘍≅𝘛𝘜𝘝, so by
CPCTC, 𝘌𝘍≅𝘜𝘝 and 𝘌𝘍=𝘜𝘝.
But, this contradicts 𝘌𝘍>𝘜𝘝.

Ex 3: Prove the Converse of the Hinge Theorem

To complete the proof of the Hinge Theorem, show that assuming 𝘮∠𝘍𝘋𝘌 < 𝘮∠𝘝𝘛𝘜 leads to a contradiction of the given statement, 𝘌𝘍>𝘜𝘝.

Ex 4: Apply the Converse of the Hinge Theorem

What are the possible values of x?

Since 𝘍𝘎<𝘊𝘋 and 𝘊𝘋<𝘈𝘉, apply the Converse of the Hinge Theorem.

Ex 4: Apply the Converse of the Hinge Theorem

What are the possible values of x?

Since 𝘍𝘎<𝘊𝘋 and 𝘊𝘋<𝘈𝘉, apply the Converse of the Hinge Theorem.

𝘮∠𝘍𝘌𝘎 <

𝘮∠𝘊𝘌𝘋 <

𝘮∠𝘈𝘊𝘉

𝖴𝗌𝖾 𝗍𝗁𝖾 𝖢𝗈𝗇𝗏𝖾𝗋𝗌𝖾 𝗈𝖿 𝗍𝗁𝖾 𝖧𝗂𝗇𝗀𝖾 𝖳𝗁𝖾𝗈𝗋𝖾𝗆.

36

2x–4

60

40

2x

64

20

x

32

The possible values for x are between 20 and 32.

Ex 4: Apply the Converse of the Hinge Theorem

What are the possible values of x?

Ex 4: Apply the Converse of the Hinge Theorem

What are the possible values of x?

𝘮inimum

<

𝘮∠(right)

<

𝘮∠(left)

0

4x–18

54

18

4x

72

4.5

x

18

The possible values for x are between 4.5 and 18.

Ex 4: Apply the Converse of the Hinge Theorem

What are the possible values of x?

Ex 4: Apply the Converse of the Hinge Theorem

What are the possible values of x?

Since 21>16,

3x+5 > 32

3x > 27

x > 9

ADDITIONAL EXAMPLE 4

Write an inequality relating m<DAC and m<BAC if DC = 36 and BC = 40.

Dropdown

If DC = 36 and BC = 40, then m∠DAC​ ​

Select answer

m∠BAC.

ADDITIONAL EXAMPLE 4

Write an inequality relating m<DAC and m<BAC if DC = 36 and BC = 40.