Indirect Proof and Inequalities 5 5 in One

Indirect Proof and Inequalities 5 -5 in One Triangle Example 2 A: Ordering Triangle

Indirect Proof and Inequalities 5 -5 in One Triangle Example 2 B: Ordering Triangle

Indirect Proof and Inequalities 5 -5 in One Triangle Check It Out! Example 2

Indirect Proof and Inequalities 5 -5 in One Triangle A triangle is formed by

Indirect Proof and Inequalities 5 -5 in One Triangle A certain relationship must exist

Indirect Proof and Inequalities 5 -5 in One Triangle Example 3 A: Applying the

Indirect Proof and Inequalities 5 -5 in One Triangle Example 3 B: Applying the

Indirect Proof and Inequalities 5 -5 in One Triangle Check It Out! Example 3

Indirect Proof and Inequalities 5 -5 in One Triangle Example 4: Finding Side Lengths

Indirect Proof and Inequalities 5 -5 in One Triangle Check It Out! Example 4

Indirect Proof and Inequalities 5 -5 in One Triangle Example 5: Travel Application The

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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. Holt Geometry

Indirect Proof and Inequalities 5 -5 in One Triangle Example 2 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 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 2 B: Ordering Triangle Side Lengths and Angle Measures Write the sides in order from shortest to longest. m R = 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 2 a Write the angles in order from smallest to largest. The shortest side is smallest angle is B. The longest side is , 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 2 b Write the sides in order from shortest to longest. m E = 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 3 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. Holt Geometry

Indirect Proof and Inequalities 5 -5 in One Triangle Example 3 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. Holt Geometry

Indirect Proof and Inequalities 5 -5 in One Triangle Check It Out! Example 3 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. Holt Geometry

Indirect Proof and Inequalities 5 -5 in One Triangle Check It Out! Example 3 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. 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