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Inequalities in One
Triangle
Warm Up
Lesson
Warm Up
1. Write a conditional from the sentence “An
isosceles triangle has two congruent sides.”
If a ∆ is isosc., then it has 2  sides.
2. Write the contrapositive of the conditional “If it
is Tuesday, then John has a piano lesson.”
If John does not have a piano lesson, then it is
not Tuesday.
3. Show that the conjecture “If x > 6, then 2x >
14” is false by finding a counterexample.
x=7
Your Goal Today is…
Recognize & apply properties of
inequalities to the measures of the angles
of a triangle.
Recognize & apply properties of
inequalities to the relationships between
the angles and sides of a triangle
Vocabulary
Inequality (page 344)
The positions of the longest and shortest sides of
a triangle are related to the positions of the
largest and smallest angles.
Example A:
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.
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
.
Your Turn! Example a
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.
Your Turn! Example 2b
Write the sides in order from
shortest to longest.
mE = 180° – (90° + 22°) = 68°
The smallest angle is D, so the shortest side is
The largest angle is F, so the longest side is
The sides from shortest to longest are
.
.
A triangle is formed by three segments, but not
every set of three segments can form a triangle.
A certain relationship must exist among the lengths
of three segments in order for them to form a
triangle.
Example A:
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.
Example B:
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.

Example C:
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
Example C Continued
Step 2 Compare the lengths.


Yes—the sum of each pair of lengths is greater
than the third length.

Your Turn! Example a
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.
Your Turn! Example b
Tell whether a triangle can have sides with the
given lengths. Explain.
6.2, 7, 9


Yes—the sum of each pair of lengths is greater
than the third side.

Your Turn! Example c
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
4t
4(4)
16
t2 + 1
(4)2 + 1
17
Example c Continued
Step 2 Compare the lengths.



Yes—the sum of each pair of lengths is greater
than the third length.
Example D:
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.
Your Turn! Example 4
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.
Example E:
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.
Your Turn! Example 5
The distance from San Marcos to Johnson City is
50 miles, and the distance from Seguin to San
Marcos is 22 miles. What is the range of
distances from Seguin to Johnson City?
Let x be the distance from Seguin to Johnson City.
x + 22 > 50
x > 28
28 < x < 72
x + 50 > 22
x > –28
22 + 50 > x
Δ Inequal. Thm.
72 > x
Subtr. Prop. of
Inequal.
Combine the inequalities.
The distance from Seguin to Johnson City is greater
than 28 miles and less than 72 miles.