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Indirect Proof and Inequalities Indirect Proof and Inequalities 5-5 5-5 in One Triangle in One Triangle Warm Up Lesson Presentation Lesson Quiz Holt Geometry Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Warm Up 1. Write a conditional from the sentence “An isosceles triangle has two congruent sides.” If a ∆ is isosc., then it has 2 sides. 2. Write the contrapositive of the conditional “If it is Tuesday, then John has a piano lesson.” If John does not have a piano lesson, then it is not Tuesday. 3. Show that the conjecture “If x > 6, then 2x > 14” is false by finding a counterexample. x=7 Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Objectives Write indirect proofs. Apply inequalities in one triangle. Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Vocabulary indirect proof Holt Geometry Indirect Proof and Inequalities 5-5 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. Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Helpful Hint When writing an indirect proof, look for a contradiction of one of the following: the given information, a definition, a postulate, or a theorem. Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 1: Writing an Indirect Proof Write an indirect proof that if a > 0, then Step 1 Identify the conjecture to be proven. Given: a > 0 Prove: Step 2 Assume the opposite of the conclusion. Assume Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 1 Continued Step 3 Use direct reasoning to lead to a contradiction. Given, opposite of conclusion Zero Prop. of Mult. Prop. of Inequality 10 However, 1 > 0. Holt Geometry Simplify. Indirect Proof and Inequalities 5-5 in One Triangle Example 1 Continued Step 4 Conclude that the original conjecture is true. The assumption that Therefore Holt Geometry is false. Indirect Proof and Inequalities 5-5 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. Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 1 Continued Step 3 Use direct reasoning to lead to a contradiction. m1 + m2 + m3 = 180° 90° + 90° + m3 = 180° 180° + m3 = 180° m3 = 0° However, by the Protractor Postulate, a triangle cannot have an angle with a measure of 0°. Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 1 Continued 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. Holt Geometry Indirect Proof and Inequalities 5-5 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. Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 2A: Ordering Triangle Side Lengths and Angle Measures Write the angles in order from smallest to largest. The shortest side is smallest angle is F. The longest side is , so the , so the largest angle is G. The angles from smallest to largest are F, H and G. Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 2B: Ordering Triangle Side Lengths and Angle Measures Write the sides in order from shortest to longest. mR = 180° – (60° + 72°) = 48° The smallest angle is R, so the shortest side is . The largest angle is Q, so the longest side is The sides from shortest to longest are Holt Geometry . Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 2a Write the angles in order from smallest to largest. The shortest side is smallest angle is B. The longest side is , so the , so the largest angle is C. The angles from smallest to largest are B, A, and C. Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 2b Write the sides in order from shortest to longest. mE = 180° – (90° + 22°) = 68° The smallest angle is D, so the shortest side is The largest angle is F, so the longest side is The sides from shortest to longest are Holt Geometry . . Indirect Proof and Inequalities 5-5 in One Triangle A triangle is formed by three segments, but not every set of three segments can form a triangle. Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle A certain relationship must exist among the lengths of three segments in order for them to form a triangle. Holt Geometry Indirect Proof and Inequalities 5-5 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. Holt Geometry Indirect Proof and Inequalities 5-5 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 Yes—the sum of each pair of lengths is greater than the third length. Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 3C: Applying the Triangle Inequality Theorem Tell whether a triangle can have sides with the given lengths. Explain. n + 6, n2 – 1, 3n, when n = 4. Step 1 Evaluate each expression when n = 4. n+6 n2 – 1 3n 4+6 (4)2 – 1 3(4) 10 15 12 Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 3C Continued Step 2 Compare the lengths. Yes—the sum of each pair of lengths is greater than the third length. Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 3a Tell whether a triangle can have sides with the given lengths. Explain. 8, 13, 21 No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths. Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 3b Tell whether a triangle can have sides with the given lengths. Explain. 6.2, 7, 9 Yes—the sum of each pair of lengths is greater than the third side. Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 3c Tell whether a triangle can have sides with the given lengths. Explain. t – 2, 4t, t2 + 1, when t = 4 Step 1 Evaluate each expression when t = 4. t–2 4–2 2 Holt Geometry 4t 4(4) 16 t2 + 1 (4)2 + 1 17 Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 3c Continued Step 2 Compare the lengths. Yes—the sum of each pair of lengths is greater than the third length. Holt Geometry Indirect Proof and Inequalities 5-5 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. x + 8 > 13 x>5 x + 13 > 8 x > –5 8 + 13 > x 21 > x Combine the inequalities. So 5 < x < 21. The length of the third side is greater than 5 inches and less than 21 inches. Holt Geometry Indirect Proof and Inequalities 5-5 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. Let x represent the length of the third side. Then apply the Triangle Inequality Theorem. x + 22 > 17 x > –5 x + 17 > 22 x>5 22 + 17 > x 39 > x Combine the inequalities. So 5 < x < 39. The length of the third side is greater than 5 inches and less than 39 inches. Holt Geometry Indirect Proof and Inequalities 5-5 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 + 51 > 46 46 + 51 > x Δ Inequal. Thm. x>5 x > –5 97 > x Subtr. Prop. of Inequal. 5 < x < 97 Combine the inequalities. The distance from San Francisco to Oakland is greater than 5 miles and less than 97 miles. Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 5 The distance from San Marcos to Johnson City is 50 miles, and the distance from Seguin to San Marcos is 22 miles. What is the range of distances from Seguin to Johnson City? Let x be the distance from Seguin to Johnson City. x + 22 > 50 x > 28 28 < x < 72 x + 50 > 22 x > –28 22 + 50 > x Δ Inequal. Thm. 72 > x Subtr. Prop. of Inequal. Combine the inequalities. The distance from Seguin to Johnson City is greater than 28 miles and less than 72 miles. Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Lesson Quiz: Part I 1. Write the angles in order from smallest to largest. C, B, A 2. Write the sides in order from shortest to longest. Holt Geometry Indirect Proof and Inequalities 5-5 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. 5 cm < x < 29 cm 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. 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. Holt Geometry