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Triangle Inequalities

- 7.1 Segments, Angles, and Inequalities

- 7.2 Exterior Angle Theorem

- 7.3 Inequalities Within a Triangle

- 7.4 Triangle Inequality Theorem

Inequalities

Segments, Angles, and Inequalities

What You'll Learn

You will learn to apply inequalities to segment

and angle measures.

1) Inequality

Segments, Angles, and Inequalities

The Comparison Property of Numbers is used to

compare two line segments ofunequal measures.

The property states that given two unequal

numbers a and b, either

a gt b

a lt b or

The same property is also used to compare angles

of unequal measures.

Segments, Angles, and Inequalities

The measure of ?J is greater than the measure of

?K.

inequalities

The statements TU gt VW and ?J gt ?K are called

__________ becausethey contain the symbol lt or

gt.

a lt b

a b

a gt b

Segments, Angles, and Inequalities

gt

SN

DN

6 (- 1)

6 2

gt

7

4

Segments, Angles, and Inequalities

AB gt CB

AB gt AC

A similar theorem for comparing angle measures is

stated below.This theorem is based on the Angle

Addition Postulate.

Segments, Angles, and Inequalities

A similar theorem for comparing angle measures is

stated below.This theorem is based on the Angle

Addition Postulate.

Segments, Angles, and Inequalities

Use theorem 7 2 to solve the following problem.

lt

m?BDA

m?CDA

45

40 45

lt

45

85

Segments, Angles, and Inequalities

For any numbers a, b, and c,

if 5 lt 8 and 8 lt 9, then 5 lt 9.

1) if a lt b and b lt c, then a lt c.

2) if a gt b and b gt c, then a gt c.

if 7 gt 6 and 6 gt 3, then 7 gt 3.

Segments, Angles, and Inequalities

For any numbers a, b, and c,

1) if a lt b, then a c lt b c and a c

lt b c.

1 lt 3 1 5 lt 3 5 6 lt 8

2) if a gt b, then a c gt b c and a c

gt b c.

For any numbers a, b, and c,

End of Section 7.1

Exterior Angle Theorem

What You'll Learn

You will learn to identify exterior angles and

remote interior angles of a triangle and use the

Exterior Angle Theorem.

1) Interior angle 2) Exterior angle 3) Remote

interior angle

Exterior Angle Theorem

interior

In the triangle below, recall that ?1, ?2, and ?3

are _______ angles of?PQR.

exterior

Angle 4 is called an _______ angle of ?PQR.

An exterior angle of a triangle is an angle that

forms a _________ with one of the angles of the

triangle.

linear pair

In ?PQR, ?4 is an exterior angle at R because it

forms a linear pair with ?3.

Remote interior angles

____________________ of a triangle are the two

angles that do not form a linear pair with the

exterior angle.

In ?PQR, ?1, and ?2 are the remote interior

angles with respect to ?4.

Exterior Angle Theorem

In the figure below, ?2 and ?3 are remote

interior angles with respect to what angle?

?5

Exterior Angle Theorem

remote interior angles

m?4

m?1 m?2

Exterior Angle Theorem

Exterior Angle Theorem

remote interior angles

m?4 gt

m?1

m?4 gt

m?2

Exterior Angle Theorem

Name two angles in the triangle below that have

measures less than 74.

?1 and ?3

acute

Exterior Angle Theorem

Exterior Angle Theorem

The feathershaped leaf is called a

pinnatifid. In the figure, does x y?

Explain.

28

__ 81 32 78

28

109 110

No! x does not equal y

End of Section 7.2

Inequalities Within a Triangle

What You'll Learn

You will learn to identify the relationships

between the _____ and _____ of a triangle.

sides

angles

Nothing New!

Inequalities Within a Triangle

in the same order

LP lt

PM lt

ML

m?M lt

m?P

m?L lt

Inequalities Within a Triangle

in the same order

m?W lt

m?K

m?J lt

JK lt

KW lt

WJ

Inequalities Within a Triangle

greatest measure

5

3

4

WY gt

XW

WY gt

XY

Inequalities Within a Triangle

The longest side is

The largest angle is

So, the largest angle is

So, the longest side is

End of Section 7.3

Triangle Inequality Theorem

What You'll Learn

You will learn to identify and use the Triangle

Inequality Theorem.

Nothing New!

Triangle Inequality Theorem

greater

a b gt c

a c gt b

b c gt a

Triangle Inequality Theorem

Can 16, 10, and 5 be the measures of the sides of

a triangle?

16 10 gt 5

No!

16 5 gt 10

End of Section 7.4

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