This document discusses relationships between sides and angles of triangles. It provides examples of ordering triangle sides by length and angles by measure. The triangle inequality theorem states that the sum of any two side lengths must be greater than the third side length for a triangle to exist. Examples show applying this theorem to determine if a triangle can exist with given side lengths, and to find the possible range of a third side given two other side lengths. The document also includes practice problems ordering triangle parts and applying the triangle inequality theorem.

1. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
Apply inequalities in one triangle.
Objectives

2. 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.

3. 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.

4. 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.
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

5. 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.

6. 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.

7. 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.

8. 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.
 


9. 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

10. 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.

11. Holt Geometry
5-5
Indirect Proof and Inequalities
in One Triangle
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

12. 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 distances
shown be 8 ft and 6 ft? Explain.
Yes; the sum of any two lengths is
greater than the third length.