5-7 Inequalities in Two Triangles

1. 5-7 Inequalities in Two Triangles

2. 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.

3. 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.

4. SAS • 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.

5. Hinge Theorem (SAS Inequality) • 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.

6. Swing Example

7. Another example • Given, • Then BC > YZ B Y Z C 1 2 A X

8. Converse of the Hinge Theorem • 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.

9. Example • Given BC > YZ, • Then B Y Z C 1 2 A X

10. What is the range of possible values for x? • 60>5x-20 • 80>5x • 16>x or x<16 R 15 S 10 60° (5x-20)° U T

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