5-5 Inequalities in One Triangle Warm Up Lesson Presentation Lesson Quiz Holt McDougal Geometry Holt Geometry

Objectives Apply 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.

Example 1A: Ordering Triangle Side Lengths and Angle Measures Write the angles in order from smallest to largest. The shortest side is , so the 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.

Check It Out! Example 1B Write the angles in order from smallest to largest. The shortest side is , so the 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.

Example 2A: Ordering Triangle Side Lengths and Angle Measures Write the sides in order from longest to shortest. mR = 180° – (60° + 72°) = 48° The largest angle is Q, so the longest side is . The smallest angle is R, so the shortest side is . The sides from longest to shortest are PR, QR, and QP

Check It Out! Example 2B Write the sides in order from longest to smallest. mE = 180° – (90° + 22°) = 68° The largest angle is F, so the longest side is . The smallest angle is D, so the shortest side is . The sides from longest to shortest are DE, DF, and EF

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 1: 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 1: 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.

Check It Out! Example 1 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.

Example 2-1: 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 2-1 Continued Step 2 Compare the lengths.    Yes—the sum of each pair of lengths is greater than the third length.

Check It Out! Example 2-2 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 4t t2 + 1 4 – 2 4(4) (4)2 + 1 2 16 17

Check It Out! Example 2-2 Continued Step 2 Compare the lengths.    Yes—the sum of each pair of lengths is greater than the third length.

Example 3A: 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 + 13 > 8 8 + 13 > x x > 5 x > –5 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.

Check It Out! Example 3B The lengths of two sides of a triangle are 6 cm and 10 cm. Find the range of possible lengths for the third side. Rule for finding the range of the third side (short cut): Maximum value- add two sides Minimum value- subtract two sides Let x represent the third side 10+6 = 16 10-6 = 4 4 < x <16 Min < x < Max Combine the inequalities. So 4 < x < 16. The length of the third side is greater than 4 cm and less than 16 cm.