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5.6Inequalities in Two Triangles Theorem 5.14: Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included.

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Presentation on theme: "5.6Inequalities in Two Triangles Theorem 5.14: Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included."— Presentation transcript:

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2 5.6Inequalities in Two Triangles Theorem 5.14: Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is _________ than the third side of the second. longer V X W R S T

3 5.6Inequalities in Two Triangles Theorem 5.15: Converse of the Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is _________ than the included angle of the second. larger A C B D E F

4 5.6Inequalities in Two Triangles Example 1 Use the Hinge Theorem and its converse Complete the statement with, or =. Explain. B C AE D F A D B C a. You are given that Solution Because 61 o < ____, by the Hinge Theorem, AC < ____.

5 5.6Inequalities in Two Triangles Example 1 Use the Hinge Theorem and its converse Complete the statement with, or =. Explain. B C AE D F A D B C b. You are given that Solution by the Reflexive Property. Because 34 > 33, ____ > ____. So, by the Converse of the Hinge Theorem, ________ > ________. and you know that

6 5.6Inequalities in Two Triangles Example 2 Solve a multi-step problem Travel Travel Car A leaves a mall, heads due north for 5 mi and then turns due west for 3 mi. Car B leaves the same mall, heads due south for 5 mi and then turns 80 o towards east for 3 mi. Which car is farther from the mall? Draw a diagram. Mall The distance driven and the distance back to the mall form two triangles, with _________ 5 mile sides and __________ 3 mile sides. congruent Add the third side to the triangle. Use linear pairs to find the included angles of _____ and _____. Because 100 o > 90 o, Car ___ is farther from the mall than Car A by the _______________. Hinge Theorem B A

7 5.6Inequalities in Two Triangles Example 3 Write an indirect proof Write an indirect proof to show that an odd number is not divisible by 6 Write an indirect proof to show that an odd number is not divisible by 6. Given x is an odd number. Prove x is not divisible by 6. Assume temporarily that ______________________. This means that ____ = n for some whole number n. x is divisible by 6 Step 1 Solution So, multiplying both sides by 6 gives ___ = ___. x6n6n

8 5.6Inequalities in Two Triangles Example 3 Write an indirect proof Write an indirect proof to show that an odd number is not divisible by 6 Write an indirect proof to show that an odd number is not divisible by 6. Given x is an odd number. Prove x is not divisible by 6. Solution If x is odd, then by definition, x cannot be divided evenly by ___. However, __ = ___ so ___ = ____ = ___. 2 Step 2 x 6n6n 3n3n We know that __ is a whole number because n is a whole number, so x can be divided evenly by __. 3n3n 2 This contradicts the given statement that _________. x is odd Therefore, the assumption that x is divisible by 6 is _____, Step 3 false which proves that _______________________. x is not divisible by 6

9 5.6Inequalities in Two Triangles Checkpoint. Complete the following exercises. 1. which is longer, AB or CB? A B D C

10 5.6Inequalities in Two Triangles Checkpoint. Complete the following exercises. In example 2, car C leaves the mall, and goes 5 miles due west then turns 85 o towards south for 3 mi. Write the cars in order from the car closest to the mall to the car farthest from the mall. Mall B A C Car A is closest Car C is next closest Car B is farthest

11 5.6Inequalities in Two Triangles Checkpoint. Complete the following exercises. 3.Suppose you want to prove the statement “If x + y = 5 and y = 2, then x = 3.” What temporary assumption could you make to prove the conclusion indirectly? You can temporarily assume that

12 5.6Inequalities in Two Triangles Pg. 305, 5.6 #1-15


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