5-7 Inequalities in Two Triangles

Earlier in this chapter, we looked at properties of individual triangles using inequalities.

We know that the largest angle is opposite the longest side.

Also, the smallest angle is opposite the shortest side.

Inequalities in Two Triangles

We have studied several ways to show that 2 triangles are congruent.

SSS, SAS, ASA, HL and AAS are the Theorems that we can use to prove one triangle congruent to a second triangle.

Inequalities in Two Triangles

The SAS Theorem told us that if 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle of another triangle, then the 2 triangles are congruent.

SAS

The Hinge (SAS Inequality) Theorem states: If two sides of one triangle are congruent to two sides of another triangle, the triangle with the larger included angle will have a larger third side.

Hinge Theorem (SAS Inequality)

Swing Example

Given,

Then BC > YZ

Another example

2

A

B

C

1

X

Y

Z

The Converse of the Hinge Theorem states: If two sides of one triangle are congruent to two sides of another triangle, and the third sides are not congruent, then the triangle with the larger third side will have a larger included angle.

Converse of the Hinge Theorem

Given BC > YZ,

Then

Example

21

A

B

C

X

Y

Z

What is the range of possible values for x?

R

60>5x-20 80>5x 16>x or x<16

U T

S15

10

60°(5x-20)°

p.336 #1-2, 6-14 all Proof #15 P.339 #26-27

Homework