1. The positions of the longest and shortest sides of a triangle are related to the positions of the largest and smallest angles.

2. Example 1A: Write the angles in order from smallest to largest. The angles from smallest to largest are F, H and G. The shortest side is, so the smallest angle is F. The longest side is, so the largest angle is G.

3. Example 1B: 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

4. 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.

5. Example 2A: 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 2B: 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.

6. Example 3C: Tell whether a triangle can have sides with the given lengths. Explain. n + 6, n 2 – 1, 3n, when n = 4. Step 1 Evaluate each expression when n = 4. n + 6 4 + 6 10 n 2 – 1 (4) 2 – 1 15 3n3n 3(4) 12 Step 2 Compare the lengths. Yes—the sum of each pair of lengths is greater than the third length.

7. 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. Combine the inequalities. So 5 < x < 21. The length of the third side is greater than 5 inches and less than 21 inches. x + 8 > 13 x > 5 x + 13 > 8 x > –5 8 + 13 > x 21 > x

8. 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 > 5 x + 51 > 46 x > –5 46 + 51 > x 97 > x 5 < x < 97 Combine the inequalities. Δ Inequal. Thm. Subtr. Prop. of Inequal. The distance from San Francisco to Oakland is greater than 5 miles and less than 97 miles.