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

Example 2A: 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.

Example 2B: 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

Check It Out! Example 2a 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.

Check It Out! Example 2b 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

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 3A: 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 3B: 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 3a 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.

Check It Out! Example 3b 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.

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.

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 + 17 > 22 22 + 17 > x x > –5 x > 5 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 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.

Let x be the distance from Seguin to Johnson City. Check It Out! 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 + 50 > 22 22 + 50 > x Δ Inequal. Thm. x > 28 x > –28 72 > x Subtr. Prop. of Inequal. 28 < x < 72 Combine the inequalities. The distance from Seguin to Johnson City is greater than 28 miles and less than 72 miles.

Lesson Quiz: Part I 1. Write the angles in order from smallest to largest. 2. Write the sides in order from shortest to longest. C, B, A

Lesson Quiz: Part II 3. The lengths of two sides of a triangle are 17 cm and 12 cm. Find the range of possible lengths for the third side. 4. Tell whether a triangle can have sides with lengths 2.7, 3.5, and 9.8. Explain. 5 cm < x < 29 cm No; 2.7 + 3.5 is not greater than 9.8. 5. Ray wants to place a chair so it is 10 ft from his television set. Can the other two distances shown be 8 ft and 6 ft? Explain. Yes; the sum of any two lengths is greater than the third length.

EXAMPLE 3 Write an indirect proof Write an indirect proof that an odd number is not divisible by 4. GIVEN : x is an odd number. PROVE : x is not divisible by 4. SOLUTION STEP 1 Assume temporarily that x is divisible by 4. This means that = n for some whole number n. So, multiplying both sides by 4 gives x = 4n. x 4

EXAMPLE 3 Write an indirect proof STEP 2 If x is odd, then, by definition, x cannot be divided evenly by 2. However, x = 4n so = = 2n. We know that 2n is a whole number because n is a whole number, so x can be divided evenly by 2. This contradicts the given statement that x is odd. x 2 4n STEP 3 Therefore, the assumption that x is divisible by 4 must be false, which proves that x is not divisible by 4.

GUIDED PRACTICE for Example 3 4. Suppose you wanted to prove the statement “If x + y = 14 and y = 5, then x = 9.” What temporary assumption could you make to prove the conclusion indirectly? How does that assumption lead to a contradiction? Assume temporarily that x = 9; since x + y = 14 and y = 5 are given, letting x = 9 leads to the contradiction 9 + 5 = 14. ANSWER

EXAMPLE 4 Prove the Converse of the Hinge Theorem Write an indirect proof of Theorem 5.14. GIVEN : AB DE BC EF AC > DF PROVE: m B > m E Proof : Assume temporarily that m B > m E. Then, it follows that either m B = m E or m B < m E.

EXAMPLE 4 Prove the Converse of the Hinge Theorem Case 1 If m B = m E, then B E. So, ABC DEF by the SAS Congruence Postulate and AC =DF. Case 2 If m B < m E, then AC < DF by the Hinge Theorem. Both conclusions contradict the given statement that AC > DF. So, the temporary assumption that m B > m E cannot be true. This proves that m B > m E.

Daily Homework Quiz 3. Suppose you want to write an indirect proof of this statement: “In ABC, if m A > 90° then ABC is not a right triangle.” What temporary assumption should start your proof? Assume ABC is a right triangle. ANSWER