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5-7 Inequalities in Two Triangles . 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.
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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.
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.
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.
Another example • Given, • Then BC > YZ B Y Z C 1 2 A X
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.
Example • Given BC > YZ, • Then B Y Z C 1 2 A X
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
Homework • p.336 #1-2, 6-14 all • Proof #15 • P.339 #26-27