Inequalities for Triangles

What we already know…

A

BC

1212

We know ∠B = ∠C

S

TU

1214

We could write a proof to show ∠T ≠∠U

*We could also prove that m∠T > m ∠U, BUT theorem 1 tells us that!

A B C

Complete each statement by writing <, =, or >.

a. AC _____ AB + BCb. AC _____ ABc. BC _____ AC

Example 1 :

Theorem 1

If one side of a triangle is longer than a second side, then the angle opposite the longer side is larger than the angle opposite the shorter side.

Example 2:

Name the largest angle and the smallest angle of the triangle.

V

U

W

9

10

11

∠U is the LARGEST angle because it is opposite the LONGEST side (WV)

∠W is the SMALLEST angle because it is opposite the SHORTEST side (VU)

Theorem 2

If one angle of a triangle is larger than a second angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.

Example 3:

Name the longest and shortest side of the triangle.

A

B

C

90º

30º

**Always find the other angle BEFORE you answer the question!!

Side AB is the LONGEST side because it is opposite the LARGEST angle (∠C)

Side CB is the SHORTEST side because it is opposite the SMALLEST angle (∠A)

Theorem 3

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

L

M

N

LM + MN > LN

MN + LN > LM

LN + LM > MN

Example 4:

The lengths of two sides of a triangle are 3 and 5. The length of the third side must be greater than _____ but less than _____.

5

3

x

Let x be the length of the third side.

x + 3 > 5

x > 2

3 + 5 > x

8 > x

x + 5 > 3

x > -2

2 8