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Inequalities in One Triangle Warm Up Lesson 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 Your Goal Today is… Recognize & apply properties of inequalities to the measures of the angles of a triangle. Recognize & apply properties of inequalities to the relationships between the angles and sides of a triangle Vocabulary Inequality (page 344) The positions of the longest and shortest sides of a triangle are related to the positions of the largest and smallest angles. Example A: 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. 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 . Your Turn! Example a 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. Your Turn! 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 . . A triangle is formed by three segments, but not every set of three segments can form a triangle. A certain relationship must exist among the lengths of three segments in order for them to form a triangle. Example A: 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. Example B: 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. Example C: 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 Example C Continued Step 2 Compare the lengths. Yes—the sum of each pair of lengths is greater than the third length. Your Turn! Example a 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. Your Turn! Example b 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. Your Turn! Example c 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 4t 4(4) 16 t2 + 1 (4)2 + 1 17 Example c Continued Step 2 Compare the lengths. Yes—the sum of each pair of lengths is greater than the third length. Example D: 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. Your Turn! 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. Example E: 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. Your Turn! 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.