Indirect Proof and Inequalities
5-5 in One Triangle
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 1: Ordering Triangle Side Lengths and Angle
Measures
Write the angles in order from
smallest to largest.
The shortest side is
smallest angle is F.
, so the
The angles from smallest to largest are F, H and G.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 2: 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 sides from shortest to longest are
Holt Geometry
48°
Indirect Proof and Inequalities
5-5 in One Triangle
Example 3:
Answer: 𝑨𝑩, 𝑩𝑪, 𝑨𝑪
Holt 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 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.
NOTE: Just check that the sum of the two
shorter sides is greater than the longest
side.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
AB + BC > AC
3x - 2 + 4x - 8 > 2x + 5
7x – 10 > 2x + 5
x>3
Holt Geometry
BC + AC > AB
4x - 8 + 2x + 5 > 3x - 2
6x – 3 > 3x - 2
𝟏
x>
𝟑
Indirect Proof and Inequalities
5-5 in One Triangle
AB + AC > BC
3x - 2 + 2x + 5 > 4x - 8
5x + 3 > 4x – 8
x > -11
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
x>3
AB + BC > AC
x>3
Holt Geometry
BC + AC > AB
x>
𝟏
𝟑
AB + AC > BC
x > -11
Indirect Proof and Inequalities
5-5 in One Triangle
Example 5: Applying the Triangle Inequality
Theorem
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 5: Applying the Triangle Inequality
Theorem
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 6: Finding Possible 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
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 Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
x + 21 > 24
21 + 24 > x
x>3
3 < x < 45
Holt Geometry
45 > x
Indirect Proof and Inequalities
5-5 in One Triangle
JK
KL
JL
5
9
4
Holt Geometry
0
-6
-6
=
=
=
25 = 5
117 ≈ 10.8
52 ≈ 7.2
∴
5 + 7.2 > 10.8
12.2 > 10.8

Indirect Proof and Inequalities
5-5 in One Triangle
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
5 + 13
≯
18
48
46
Holt Geometry