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Chapter 5 Section 5.5

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Chapter 5Section 5.5

Inequalities in a triangle

-3

-3

+9

-2x

-10

-10

+9

-2x

-1

2

-1

2

Don’t forget to change it!

3 < 2x - 9

x < 11

2 > -x

12 < 2x

-2<x

6<x

Compare Sides to Order Angles

C

B

A

Theorem

Theorem 5.10

If one side of a triangle is longer than another side,

then the angle opposite the longer side is larger than

the angle opposite the shorter side.

mA > mB > mC

CB > CA > AB

Compare Angles to Order Sides

R

P

Q

Theorem

Theorem 5.11

If one Angle of a triangle is larger than another angle,

then the side opposite the larger angle is longer than

the side opposite the smaller angle.

mP > mQ > mR

QR > PR > PQ

mM > mK > mL

KL > LM > KM

mM > mO > mN

ON > MN > OM

mB > mA > mC

AC > BC> AB

EG > GF > EF

mF > mE > mG

mJ > mI > mH

HI > HJ > IJ

LM > KM > KL

mK > mL > mM

In ABC

CB > AB > AC

In CBD

BD > CD > CB

Thus

BD > CD > CB > AB > AC

In LKM

LM > KL > KM

In LMN

MN > LN > LM

Thus

MN > LN > LM > LN > LM

Inequalities in a triangle

R

P

Q

T

Theorem

Theorem 5.12 Exterior Angle Inequality

The measure of an exterior angle of a triangle is greater

than the measure of the two nonadjacent interior angles.

TQR is an exterior angle of QRP

mTQR > mR

And

mTQR > mP

Inequalities in a triangle

Theorem

Theorem 5.13 Triangle Inequality

The sum of the lengths of any two sides of a triangle

is greater than the length of the third side.

AB + BC > AC

AB + AC > BC

BC + AC > AB

The sum of any two sides must be greater than the third side!

YZ + XY > XZ

XZ + XY > YZ

XZ + YZ > XY

2 + XY > 3

3 + XY > 2

2 + 3 > XY

XY > 1

5 > XY

XY > -1

5 > XY > 1

The sum of any two sides must be greater than the third side!

YZ + XY > XZ

XZ + XY > YZ

XZ + YZ > XY

8 + XY > 10

10 + XY > 8

8 + 10 > XY

XY > 2

18 > XY

XY > -2

18 > XY > 2

CC

60

AF

40

DS

60 + x > 40

60 + 40 > x

x + 40 > 60

100 > x

x > -20

x > 20

x

100 > x > 20

No, the distance must be less than 100 miles