Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle5-5 Indirect Proof and Inequalities

in One Triangle

Holt Geometry

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Warm Up

1. Write a conditional from the sentence “An isosceles triangle has two congruent sides.”

2. Write the contrapositive of the conditional “If it is Tuesday, then John has a piano lesson.”

3. Show that the conjecture “If x > 6, then 2x > 14” is false by finding a counterexample.

If a ∆ is isosc., then it has 2 sides.

If John does not have a piano lesson, then it is not Tuesday.

x = 7

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Apply inequalities in one triangle.

Learning Targets

Holt Geometry

5-5 Indirect Proof and 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.

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Example 2A: Ordering Triangle Side Lengths and Angle Measures

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.

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Example 2B: Ordering Triangle Side Lengths and Angle Measures

Write the sides in order from shortest to longest.

mR = 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

5-5 Indirect Proof and Inequalities in One Triangle

A triangle is formed by three segments, but not every set of three segments can form a triangle.

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

A certain relationship must exist among the lengths of three segments in order for them to form a triangle.

Holt Geometry

5-5 Indirect Proof and Inequalities in One 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.

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

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.

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Example 3C: 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

4 + 6

10

n2 – 1

(4)2 – 1

15

3n

3(4)

12

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Example 3C Continued

Step 2 Compare the lengths.

Yes—the sum of each pair of lengths is greater than the third length.

Holt Geometry

5-5 Indirect Proof and Inequalities 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.

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

Holt Geometry

5-5 Indirect Proof and Inequalities 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.

Combine the inequalities. So 5 < x < 39. The length of the third side is greater than 5 inches and less than 39 inches.

x + 22 > 17

x > –5

x + 17 > 22

x > 5

22 + 17 > x

39 > x

Holt Geometry

5-5 Indirect Proof and Inequalities 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 > 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.

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

Lesson Quiz: Part I1. Write the angles in order from smallest to

largest.

2. Write the sides in order from shortest to longest.

C, B, A

Holt Geometry

5-5 Indirect Proof and Inequalities in One Triangle

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.

No; 2.7 + 3.5 is not greater than 9.8.

5 cm < x < 29 cm

5. Ray wants to place a chair so it is 10 ft from his television set. Can the other two distancesshown be 8 ft and 6 ft? Explain.

Yes; the sum of any two lengths is greater than the third length.