1. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle 5-5 Indirect Proof and Inequalities in One Triangle Holt Geometry

2. 5-5 Indirect Proof and Inequalities in One Triangle Warm Up 1. Write a conditional from the sentence “An isosceles triangle has two congruent sides.” 2. Write the contrapositive of the conditional “If it is Tuesday, then John has a piano lesson.” 3. Show that the conjecture “If x > 6, then 2x > 14” is false by finding a counterexample.

3. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle Write indirect proofs. Apply inequalities in one triangle. Objectives

4. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle indirect proof Vocabulary

5. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle So far you have written proofs using direct reasoning. You began with a true hypothesis and built a logical argument to show that a conclusion was true. In an indirect proof, you begin by assuming that the conclusion is false. Then you show that this assumption leads to a contradiction. This type of proof is also called a proof by contradiction.

6. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle

7. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle When writing an indirect proof, look for a contradiction of one of the following: the given information, a definition, a postulate, or a theorem. Helpful Hint

8. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle Example 1: Writing an Indirect Proof Step 1 Identify the conjecture to be proven. Given: a > 0 Step 2 Assume the opposite of the conclusion. Write an indirect proof that if a > 0, then Prove:Assume

9. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle Example 1 Continued Step 3 Use direct reasoning to lead to a contradiction. However, 1 > 0. 1  0 Given, opposite of conclusion Zero Prop. of Mult. Prop. of Inequality Simplify.

10. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle Step 4 Conclude that the original conjecture is true. Example 1 Continued The assumption that is false. Therefore

11. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle Check It Out! Example 1 Write an indirect proof that a triangle cannot have two right angles. Step 1 Identify the conjecture to be proven. Given: A triangle’s interior angles add up to 180°. Prove: A triangle cannot have two right angles. Step 2 Assume the opposite of the conclusion. An angle has two right angles.

12. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle Check It Out! Example 1 Continued Step 3 Use direct reasoning to lead to a contradiction. However, by the Protractor Postulate, a triangle cannot have an angle with a measure of 0°. m1 + m2 + m3 = 180° 90° + 90° + m3 = 180° 180° + m3 = 180° m3 = 0°

13. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle Step 4 Conclude that the original conjecture is true. The assumption that a triangle can have two right angles is false. Therefore a triangle cannot have two right angles. Check It Out! Example 1 Continued

14. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle The positions of the longest and shortest sides of a triangle are related to the positions of the largest and smallest angles.

15. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle Example 2A: Ordering Triangle Side Lengths and Angle Measures Write the angles in order from smallest to largest.

16. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle Example 2B: Ordering Triangle Side Lengths and Angle Measures Write the sides in order from shortest to longest.

17. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle A triangle is formed by three segments, but not every set of three segments can form a triangle.

18. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle A certain relationship must exist among the lengths of three segments in order for them to form a triangle.

19. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle Example 3A: Applying the Triangle Inequality Theorem Tell whether a triangle can have sides with the given lengths. Explain. 7, 10, 19 No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths.

20. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle Example 3B: Applying the Triangle Inequality Theorem Tell whether a triangle can have sides with the given lengths. Explain. 2.3, 3.1, 4.6

21. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle Check It Out! Example 3a Tell whether a triangle can have sides with the given lengths. Explain. 8, 13, 21

22. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle Example 4: Finding Side Lengths The lengths of two sides of a triangle are 8 inches and 13 inches. Find the range of possible lengths for the third side. Let x represent the length of the third side. Then apply the Triangle Inequality Theorem. Combine the inequalities. So 5 < x < 21. The length of the third side is greater than 5 inches and less than 21 inches. x + 8 > 13 x > 5 x + 13 > 8 x > –5 8 + 13 > x 21 > x

23. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle Check It Out! Example 4 The lengths of two sides of a triangle are 22 inches and 17 inches. Find the range of possible lengths for the third side.

24. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle Example 5: Travel Application The figure shows the approximate distances between cities in California. What is the range of distances from San Francisco to Oakland? Let x be the distance from San Francisco to Oakland. x + 46 > 51 x > 5 x + 51 > 46 x > –5 46 + 51 > x 97 > x 5 < x < 97 Combine the inequalities. Δ Inequal. Thm. Subtr. Prop. of Inequal. The distance from San Francisco to Oakland is greater than 5 miles and less than 97 miles.

25. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle Lesson Quiz: Part I 1. Write the angles in order from smallest to largest. 2. Write the sides in order from shortest to longest. C, B, A

26. Holt Geometry 5-5 Indirect Proof and Inequalities in One Triangle Lesson Quiz: Part II 3. The lengths of two sides of a triangle are 17 cm and 12 cm. Find the range of possible lengths for the third side. 4. Tell whether a triangle can have sides with lengths 2.7, 3.5, and 9.8. Explain. No; 2.7 + 3.5 is not greater than 9.8. 5 cm < x < 29 cm 5. Ray wants to place a chair so it is 10 ft from his television set. Can the other two distances shown be 8 ft and 6 ft? Explain. Yes; the sum of any two lengths is greater than the third length.