GEOMETRY

4-6 Triangle Inequalities

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

GEOMETRY

4-6 Triangle Inequalities

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

GEOMETRY

4-6 Triangle Inequalities

Apply inequalities in one triangle.

Objectives

GEOMETRY

4-6 Triangle Inequalities

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

GEOMETRY

4-6 Triangle Inequalities

Example 1: Ordering Triangle 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.

GEOMETRY

4-6 Triangle Inequalities

TEACH! Example 1

Write the angles in order from smallest to largest.

The angles from smallest to largest are B, A, and C.

The shortest side is , so the smallest angle is B.

The longest side is , so the largest angle is C.

GEOMETRY

4-6 Triangle Inequalities

Example 2: Ordering Triangle Side Lengths 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

GEOMETRY

4-6 Triangle Inequalities

TEACH! Example 2

Write the sides in order from shortest to longest.

mE = 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

GEOMETRY

4-6 Triangle Inequalities

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

GEOMETRY

4-6 Triangle Inequalities

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

GEOMETRY

4-6 Triangle Inequalities

Example 3: 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.

GEOMETRY

4-6 Triangle Inequalities

TEACH! Example 3

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.

GEOMETRY

4-6 Triangle Inequalities

TEACH! Example 4

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.

GEOMETRY

4-6 Triangle Inequalities

Example 5: 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

GEOMETRY

4-6 Triangle Inequalities

Example 5 Continued

Step 2 Compare the lengths.

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

GEOMETRY

4-6 Triangle Inequalities

TEACH! Example 5

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

4 – 2

2

t2 + 1

(4)2 + 1

17

4t

4(4)

16

GEOMETRY

4-6 Triangle Inequalities

TEACH! Example 5 Continued

Step 2 Compare the lengths.

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

GEOMETRY

4-6 Triangle Inequalities

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

GEOMETRY

4-6 Triangle Inequalities

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

GEOMETRY

4-6 Triangle Inequalities

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.

GEOMETRY

4-6 Triangle Inequalities

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 > 28

x + 50 > 22

x > –28

22 + 50 > x

72 > x

28 < x < 72 Combine the inequalities.

Δ Inequal. Thm.

Subtr. Prop. of Inequal.

The distance from Seguin to Johnson City is greater than 28 miles and less than 72 miles.

GEOMETRY

4-6 Triangle Inequalities

Lesson-4 Quiz: Part I

Use the diagram for Items 1–3. Find each measure.

1. ED

2. AB

3. mBFE

10

14

44°

GEOMETRY

4-6 Triangle Inequalities

Lesson-4 Quiz: Part II

4. Find the value of n.

5. ∆XYZ is the midsegment triangle of ∆WUV.

What is the perimeter of ∆XYZ?

16

11.5

GEOMETRY

4-6 Triangle Inequalities

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

GEOMETRY

4-6 Triangle Inequalities

Lesson Quiz: Part II3. 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.