Obj. 22 Triangle Inequalities

The student is able to (I can):

• Analyze the relationship between the angles of a triangle and the lengths of the sides

• Determine allowable lengths for sides of triangles

Thm 5-5-1

Thm 5-5-2

If two sides of a triangle are not congruent, then the larger angle is opposite the longer side.

If two angles of a triangle are not congruent, then the longer side is opposite the larger angle.

A

C

T

AT > AC → m∠C > m∠T

m∠C > m∠T → AT > AC

Triangle Inequality Theorem

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

Example:

1. Which set of lengths forms a triangle?

4, 5, 10 7, 9, 12

4 + 5 < 10 � 7 + 9 > 12 �

Note: To find a range of possible third sides given two sides, subtract for the lower bound and add for the upper bound.

Examples:

2. What is a possible third side for a triangle with sides 8 and 14?

14 — 8 = 6 lower bound

14 + 8 = 22 upper bound

The third side can be between 6 and 22.

3. What is the range of values for the third side of a triangle with sides 11 and 19?

19 — 11 = 8 lower bound

19 + 11 = 30 upper bound

8 < x < 30

Hinge Theorem and Converse

If two sides of one triangle are congruent to two sides of another triangle …

• If the included angles are not congruent, then the longer third side is across from the larger included angle.

• If the third sides are not congruent, then the larger included angle is across from the longer third side.

A

B CFE

D

∠∠ > ⇔ >Em mB AC DF

Example: Find the range of values for x.

1.

2.

(2x+8)°

26°

7

8

< + <

− < <

− < <

0 2x 8 26

8 2x 18

4 x 9

64°

35°

x+7 2x−5

< − − < +0 2x 5 and 2x 5 x 7

<

<

5 2x

2.5 x

− <

<

x 5 7

x 12

< <2.5 x 12