5.5 Inequalities in Triangles

Chapter 5

Relationships Within

Triangles

5.5 Inequalities in Triangles

Corollary to the Triangle Exterior Angle Theorem:

The measure of an exterior angle of a triangle

is greater than the measure of each of its

remote interior angles.

2

3

1

m<1 > m<2 and m<1 > m<3

Applying the Corollary

m<2 = m<1 by the Isosceles Triangle Theorem.

Explain why m<2 > m>3.

2

1

4

3

m<1 > m<3 + m<4 because

<1 is the exterior angle, so

m<1 > m<3.

By substitution property, m< 2 > m<3,

since m<2 = m<1.

Theorem 5-10

If two sides of a triangle are not congruent, then

the larger angle lies opposite the longer side.

X

Y

Z 14

12 11

<Y is the largest angle.

Comparing Angles

A landscape architect is designing a triangular

deck. She wants to place benches in the two

larger corners. Which corners have the larger

angles?

21ft

27ft

18ft

A

B

C

<B and <A are the larger angles, <C is the smallest.

Theorem 5-11

If two sides of a triangle are not congruent, then

the longer side lies opposite the larger angle.

X

Y

Z

48

98

34

XZ is the longest side.

Using Theorem 5-11

Which side is the shortest?

52 62

66

U

T

V

TV is the shortest side.

40 60

X

Y

Z

80

YZ is the shortest side

Theorem 5-12

Triangle Inequality Theorem:

The sum of the lengths of any two sides of a

triangle is greater than the length of the third

side.

a

b

c a + b > c

b + c > a

c + a > b

Triangle Inequality Theorem

Can a triangle have sides with the given lengths?

3ft, 7ft, 8ft

3cm, 6cm, 10cm

Yes, 3 + 7 = 10 and 10 > 8

No, 3 + 6 = 9 and 9 is not greater than 10

Triangle Inequality Theorem

Can a triangle have sides with the given lengths?

2m, 7m, 9m

4yd, 6yd, 9yd

No, 2 + 7 = 9, and 9 is not greater than 9

Yes, 4 + 6 = 10 and 10 is greater than 9

Finding Possible Side Lengths

A triangle has side lengths of 8cm and 10cm.

Describe the possible lengths of the third side.

The value of the third side must be greater

Than 2 and less than 18.

(x > 2 and x < 18)

2cm < x < 18cm

To answer this kind of question, add the numbers together and

Subtract the small number from the larger number.

8 + 10 = 18 10 – 8 = 2

Finding Possible Side Lengths

A triangle has side lengths of 3in and 12in.

Describe the possible lengths of the third side.

9in < x < 15in

To answer this kind of question, add the numbers together and

Subtract the small number from the larger number.

3 + 12 = 15 12 – 3 = 9

The value of the third side must be greater

Than 9 and less than 15. (x > 9 and x < 15)

Practice