1. 5-5 Inequalities in One Triangle Warm Up Lesson Presentation
Lesson Quiz
Holt McDougal Geometry
Holt Geometry

2. Objectives
Apply inequalities in one triangle.

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

4. Example 1A: 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.

5. Check It Out! Example 1B
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.

6. Example 2A: Ordering Triangle Side Lengths and Angle Measures
Write the sides in order from longest to shortest.
mR = 180° – (60° + 72°) = 48°
The largest angle is Q, so the longest side is .
The smallest angle is R, so the shortest side is .
The sides from longest to shortest are PR, QR, and QP

7. Check It Out! Example 2B
Write the sides in order from longest to smallest.
mE = 180° – (90° + 22°) = 68°
The largest angle is F, so the longest side is .
The smallest angle is D, so the shortest side is .
The sides from longest to shortest are DE, DF, and EF

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

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

10. Example 1: 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.

11. Example 1: 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.

12. Check It Out! Example 1
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.

13. Example 2-1: 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
n2 – 1 3n
4 + 6
(4)2 – 1
3(4) 10 15 12

14. Example 2-1 Continued
Step 2 Compare the lengths.   
Yes—the sum of each pair of lengths is greater than the third length.

15. Check It Out! Example 2-2
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 4t
t2 + 1
4 – 2
4(4)
(4)2 + 1 2 16 17

16. Check It Out! Example 2-2 Continued
Step 2 Compare the lengths.   
Yes—the sum of each pair of lengths is greater than the third length.

17. Example 3A: 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
> 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.

18. Check It Out! Example 3B
The lengths of two sides of a triangle are 6 cm and 10 cm. Find the range of possible lengths for the third side.
Rule for finding the range of the third side (short cut): Maximum value- add two sides
Minimum value- subtract two sides
Let x represent the third side
10+6 = 16
10-6 = 4
4 < x <16
Min < x < Max
Combine the inequalities. So 4 < x < 16. The length of the third side is greater than 4 cm and less than 16 cm.