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Indirect Proof and Inequalities 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. Show that the conjecture “If x > 6, then 2x > 14” is false by finding a counterexample. x=7 2. Solve and graph the inequality. -2x+7>10 or 4x-11>21 3 x>3 or x>8 8 3. Is 2 a solution to the inequality above? no 4. Is 4 a solution to the inequality above? yes Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Objectives Apply inequalities in one triangle. 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. Smallest side Middle angle Largest side Largest angle Smallest angle Medium side Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle What are some inequalites we could write for this triangle, comparing the sides? comparing the angles? A AB<AC AC>BC B Holt Geometry Angle A < Angle B C Indirect Proof and Inequalities 5-5 in One Triangle Example 1: Ordering Triangle Side Lengths and Angle Measures Write the angles in order from smallest to largest. The angles from smallest to largest are F, H, G Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 2: Ordering Triangle Side Lengths and Angle Measures Write the sides in order from shortest to longest. mR = 180° – (60° + 72°) = 48° The sides from shortest to longest are Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 3 Write the angles in order from smallest to largest. The angles from smallest to largest are B, A, C Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 4 Write the sides in order from shortest to longest. mE = 180° – (90° + 22°) = 68° 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. **The smallest two sides must be bigger than largest side.** Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 5: 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 6: 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 the smaller sides is greater than the third side length. Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 7: 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 the smaller sides is greater than the third side length. Holt Geometry Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 8 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 Example 9: 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 10 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 11: 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 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