Indirect Proof and Inequalities
5-5
in One Triangle
5-5 inInequalities
One Triangle
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
GeometryGeometry
Holt
Indirect Proof and Inequalities
5-5 in One Triangle
Objectives
Apply inequalities in one triangle.
Holt McDougal Geometry
Indirect Proof and Inequalities
5-5 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 McDougal Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 1A: Ordering Triangle Side Lengths and
Angle Measures
Write the angles in order from
smallest to largest.
The shortest side is
smallest angle is F.
The longest side is
, so the
, so the largest angle is G.
The angles from smallest to largest are F, H and G.
Holt McDougal Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Check It Out! Example 1B
Write the angles in order from
smallest to largest.
The shortest side is
smallest angle is B.
The longest side is
, so the
, so the largest angle is C.
The angles from smallest to largest are B, A, and C.
Holt McDougal Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
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
Holt McDougal Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
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
Holt McDougal Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
A triangle is formed by three segments, but not
every set of three segments can form a triangle.
Holt McDougal Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
A certain relationship must exist among the lengths
of three segments in order for them to form a
triangle.
Holt McDougal Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
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.
Holt McDougal Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
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.
Holt McDougal Geometry

Indirect Proof and Inequalities
5-5 in One Triangle
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.
Holt McDougal Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
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
Holt McDougal Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 2-1 Continued
Step 2 Compare the lengths.


Yes—the sum of each pair of lengths is greater
than the third length.
Holt McDougal Geometry

Indirect Proof and Inequalities
5-5 in One Triangle
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
4–2
2
Holt McDougal Geometry
4t
4(4)
16
t2 + 1
(4)2 + 1
17
Indirect Proof and Inequalities
5-5 in One Triangle
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.
Holt McDougal Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
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>5
x + 13 > 8
x > –5
8 + 13 > x
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.
Holt McDougal Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
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
10+6 = 16 10-6 = 4
third side
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.
Holt McDougal Geometry