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Indirect
Proof and Inequalities
Inequalities
5-5
5-5 in One Triangle
in One Triangle
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Geometry
Holt
Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Warm Up
1. Show that the conjecture “If x > 6, then 2x > 14” is false
by finding a counterexample.
x=7
2. Solve and graph the inequality.
-2x+7>10 or 4x-11>21
3
x>3 or x>8
8
3. Is 2 a solution to the inequality above?
no
4. Is 4 a solution to the inequality above?
yes
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Objectives
Apply inequalities in one triangle.
Holt 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.
Smallest
side
Middle
angle
Largest
side
Largest
angle
Smallest
angle
Medium
side
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
What are some inequalites we could write for
this triangle, comparing the sides? comparing
the angles?
A
AB<AC
AC>BC
B
Holt Geometry
Angle A < Angle B
C
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 angles from smallest to largest are
F, H, 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 sides from shortest to longest are
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Check It Out! Example 3
Write the angles in order from
smallest to largest.
The angles from smallest to largest are
B, A, C
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Check It Out! Example 4
Write the sides in order from
shortest to longest.
mE = 180° – (90° + 22°) =
68°
The sides from shortest to longest are
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.
**The smallest two sides must be bigger than
largest side.**
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 5: 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 Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 6: 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 the smaller sides is greater than
the third side length.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 7: 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 Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 3C Continued
Step 2 Compare the lengths.

Yes—the sum of the smaller sides is greater than
the third side length.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Check It Out! Example 8
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 Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 9: 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 Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Check It Out! Example 10
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.
x + 22 > 17
x > –5
x + 17 > 22
x>5
22 + 17 > x
39 > x
Combine the inequalities. So 5 < x < 39. The length
of the third side is greater than 5 inches and less
than 39 inches.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 11: 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 + 51 > 46 46 + 51 > x Δ Inequal. Thm.
x>5
x > –5
97 > x
Subtr. Prop. of
Inequal.
5 < x < 97 Combine the inequalities.
The distance from San Francisco to Oakland is
greater than 5 miles and less than 97 miles.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Lesson Quiz: Part I
1. Write the angles in order from smallest to
largest.
C, B, A
2. Write the sides in order from shortest to
longest.
Holt Geometry
Indirect Proof and Inequalities
5-5 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.
5 cm < x < 29 cm
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. 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.
Holt Geometry