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Example 1A:

The positions of the longest and shortest sides of a triangle are related to the positions of the largest and smallest angles. The shortest side is , so the smallest angle is  F. The longest side is , so the largest angle is  G. Example 1A:.

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Example 1A:

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

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

  3. 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 Example 1B: Write the sides in order from shortest to longest. mR = 180° – (60° + 72°) = 48°

  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. Tell whether a triangle can have sides with the given lengths. Explain. Example 2B: 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, 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 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. 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.

  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 + 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.

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