Chapter 5Section 5.5

Inequalities in a triangle

-3 -3

x < 11

-10 -10

2 > -x-1 -1

-2 < x

Don’t forget to change it!

-2x -2x

3 < 2x - 9+9 +9

12 < 2x2 2

6 < x

Compare Sides to Order Angles

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.

A B

CCB > CA > AB mA > mB > mC

Compare Angles to Order Sides

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.

Q P

R

QR > PR > PQ

mP > mQ > mR

mM > mK > mL

KL > LM > KM

mM > mO > mN

ON > MN > OM

mB > mA > mC AC > BC> AB

mF > mE > mG

EG > GF > EF

mJ > mI > mHHI > HJ > IJ

mK > mL > mMLM > KM > KL

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

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.

mTQR > mR

TQR is an exterior angle of QRP

Inequalities in a triangle

Q P

R

T

And

mTQR > mP

Theorem

Theorem 5.13 Triangle Inequality

The sum of the lengths of any two sides of a triangleis greater than the length of the third side.

AB + BC > AC

AB + AC > BC

BC + AC > AB

Inequalities in a triangle

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

XZ + YZ > XY XZ + XY > YZ YZ + XY > XZ

2 + 3 > XY

5 > XY

2 + XY > 3

XY > 1

3 + XY > 2

XY > -1

5 > XY > 1

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

XZ + YZ > XY XZ + XY > YZ YZ + XY > XZ

8 + 10 > XY

18 > XY

8 + XY > 10

XY > 2

10 + XY > 8

XY > -2

18 > XY > 2

CC

AF

DS

60

40

x

60 + 40 > x 60 + x > 40100 > x x > -20

x + 40 > 60

x > 20

100 > x > 20

No, the distance must be less than 100 miles