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Triangle Inequalities Triangle Inequalities § 7.1 Segments, Angles, and Inequalities 7.1 Segments, Angles, and Inequalities § § 7.4 Triangle Inequality Theorem 7.4 Triangle Inequality Theorem § 7.3 Inequalities Within a Triangle 7.3 Inequalities Within a Triangle § 7.2 Exterior Angle Theorem 7.2 Exterior Angle Theorem

# § 7.1 Segments, Angles, and Inequalities § 7.4 Triangle Inequality Theorem § 7.3 Inequalities Within a Triangle § 7.2 Exterior Angle Theorem

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Triangle InequalitiesTriangle InequalitiesTriangle InequalitiesTriangle Inequalities

§§ 7.1 Segments, Angles, and Inequalities 7.1 Segments, Angles, and Inequalities

§§ 7.4 Triangle Inequality Theorem 7.4 Triangle Inequality Theorem

§§ 7.3 Inequalities Within a Triangle 7.3 Inequalities Within a Triangle

§§ 7.2 Exterior Angle Theorem 7.2 Exterior Angle Theorem

Segments, Angles, and InequalitiesSegments, Angles, and Inequalities

You will learn to apply inequalities to segment and angle measures.

1) Inequality

Inequalities

Segments, Angles, and InequalitiesSegments, 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 < b or a > b

The same property is also used to compare angles of unequal measures.

T U2 cm

V W4 cm

The length of is less than the length of , or TU < VW TU VW

Segments, Angles, and InequalitiesSegments, Angles, and Inequalities

J

133°K

60°

The measure of J is greater than the measure of K.

The statements TU > VW and J > K are called __________ becausethey contain the symbol < or >.

inequalities

Postulate 7 – 1

ComparisonProperty

For any two real numbers, a and b, exactly one of the following statements is true.

a < b a = b a > b

Segments, Angles, and InequalitiesSegments, Angles, and Inequalities

6420-2

S D N

Replace with <, >, or = to make a true statement.

SN DN

6 – (- 1) 6 – 2

7 4>

>

Lesson 2-1Finding Distanceon a number line.

Segments, Angles, and InequalitiesSegments, Angles, and Inequalities

Theorem7 – 1

If point C is between points A and B, and A, C, and B are collinear, then ________ and ________.

A C B

AB > AC AB > CB

A similar theorem for comparing angle measures is stated below.This theorem is based on the Angle Addition Postulate.

Segments, Angles, and InequalitiesSegments, Angles, and Inequalities

Theorem7 – 2

then,EF and EDbetween is EP If

and DEPm DEFm PEFm DEFm

D

P

F

E

A similar theorem for comparing angle measures is stated below.This theorem is based on the Angle Addition Postulate.

Segments, Angles, and InequalitiesSegments, Angles, and Inequalities

108°

149°45°

40°

18°

A

B

C

D

Replace with <, >, or = to make a true statement.

mBDA mCDA

45° 40° + 45°

<

<

Use theorem 7 – 2 to solve the following problem.

CDAm BDA m

then,DA and DCbetween is DB Since

Check:

CDA BDA mm

45° 85°

Segments, Angles, and InequalitiesSegments, Angles, and Inequalities

Property

TransitiveProperty

For any numbers a, b, and c,

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

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

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

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

Segments, Angles, and InequalitiesSegments, Angles, and Inequalities

Property

Multiplicationand DivisionProperties

For any numbers a, b, and c,

For any numbers a, b, and c,

1) if a < b, then a + c < b + c and a – c < b – c.

2) if a > b, then a + c > b + c and a – c > b – c.

1 < 31 + 5 < 3 + 5

6 < 8

c

b

c

a andbc ac

then b, a and 0 c If )

1

c

b

c

a andbc ac

then b, a and 0 c If )

2 36 24

2 18 2 12

18 12

962

18

2

12

18 12

Exterior Angle TheoremExterior Angle Theorem

You will learn to identify exterior angles and remote interiorangles of a triangle and use the Exterior Angle Theorem.

1) Interior angle

2) Exterior angle

3) Remote interior angle

Exterior Angle Theorem Exterior Angle Theorem

1

2 3 4

P

Q R

In the triangle below, recall that 1, 2, and 3 are _______ angles ofΔPQR.

interior

Angle 4 is called an _______ angle of ΔPQR.exterior

An exterior angle of a triangle is an angle that forms a _________ with one ofthe angles of the triangle.

linear pair

In ΔPQR, 4 is an exterior angle at R because it forms a linear pair with 3.

____________________ of a triangle are the two angles that do not forma linear pair with the exterior angle.Remote interior angles

In ΔPQR, 1, and 2 are the remote interior angles with respect to 4.

Exterior Angle Theorem Exterior Angle Theorem

1

2

3 4 5

In the figure below, 2 and 3 are remote interior angles with respect towhat angle? 5

Exterior Angle Theorem Exterior Angle Theorem

Theorem 7 – 3

Exterior Angle

Theorem

The measure of an exterior angle of a triangle is equal to sum

of the measures of its ___________________.remote interior angles

X

432

1

ZY

m4 = m1 + m2

Exterior Angle Theorem Exterior Angle Theorem

Exterior Angle Theorem Exterior Angle Theorem

Theorem 7 – 4

Exterior Angle

InequalityTheorem

The measure of an exterior angle of a triangle is greater than the measures of either of its two ____________________.remote interior angles

X

432

1

ZY

m4 > m1m4 > m2

Exterior Angle Theorem Exterior Angle Theorem

1 and 3

74°

1 3

2

Name two angles in the triangle below that have measures less than 74°.

Theorem 7 – 5If a triangle has one right angle, then the other two angles

must be _____.acute

Exterior Angle Theorem Exterior Angle Theorem

3 and 1

Exterior Angle Theorem Exterior Angle Theorem

The feather–shaped leaf is called a pinnatifid.In the figure, does x = y? Explain.

x = y?

__ + 81 = 32 + 7828

28°

109 = 110

No! x does not equal y

Inequalities Within a Triangle Inequalities Within a Triangle

You will learn to identify the relationships between the _____and _____ of a triangle.

sidesangles

Nothing New!

Inequalities Within a Triangle Inequalities Within a Triangle

Theorem 7 – 6

If the measures of three sides of a triangle are unequal,

then the measures of the angles opposite those sides

are unequal ________________.

13

811

L

P

M

in the same order

LP < PM < ML

mM < mPmL <

Inequalities Within a Triangle Inequalities Within a Triangle

Theorem 7 – 7

If the measures of three angles of a triangle are unequal,

then the measures of the sides opposite those angles

are unequal ________________.in the same order

JK < KW < WJ

mW < mKmJ <

J

45°W K

60°

75°

Inequalities Within a Triangle Inequalities Within a Triangle

Theorem 7 – 8

In a right triangle, the hypotenuse is the side with the

________________.greatest measure

WY > XW

35

4 Y

W

X

WY > XY

Inequalities Within a Triangle Inequalities Within a Triangle

A

The longest side is BC

So, the largest angle is

LThe largest angle is

MNSo, the longest side is

Triangle Inequality Theorem Triangle Inequality Theorem

You will learn to identify and use the Triangle Inequality Theorem.

Nothing New!

Triangle Inequality Theorem Triangle Inequality Theorem

Theorem 7 – 9

TriangleInequalityTheorem

The sum of the measures of any two sides of a triangle is

_______ than the measure of the third side.greater

a

b

c

a + b > c

a + c > b

b + c > a

Triangle Inequality Theorem Triangle Inequality Theorem

Can 16, 10, and 5 be the measures of the sides of a triangle?

No! 16 + 10 > 5

16 + 5 > 10

However, 10 + 5 > 16

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