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4-6 Triangle Inequalities
Warm Up
Lesson Presentation
Lesson Quiz
GEOMETRY
4-6 Triangle Inequalities
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
GEOMETRY
4-6 Triangle Inequalities
Objectives
Apply inequalities in one triangle.
GEOMETRY
4-6 Triangle Inequalities
The positions of the longest and shortest
sides of a triangle are related to the positions of the
largest and smallest angles.
GEOMETRY
4-6 Triangle Inequalities
Example 1: Ordering Triangle 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.
GEOMETRY
4-6 Triangle Inequalities
TEACH! Example 1
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.
GEOMETRY
4-6 Triangle Inequalities
Example 2: Ordering Triangle Side Lengths 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
GEOMETRY
4-6 Triangle Inequalities
TEACH! Example 2
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
GEOMETRY
4-6 Triangle Inequalities
A triangle is formed by three segments, but not
every set of three segments can form a triangle.
GEOMETRY
4-6 Triangle Inequalities
A certain relationship must exist among the lengths
of three segments in order for them to form a
triangle.
GEOMETRY
4-6 Triangle Inequalities
Example 3:
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.
GEOMETRY
4-6 Triangle Inequalities
TEACH! Example 3
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.
GEOMETRY
4-6 Triangle Inequalities
TEACH! Example 4
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.
GEOMETRY
4-6 Triangle Inequalities
Example 5:
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
GEOMETRY
4-6 Triangle Inequalities
Example 5 Continued
Step 2 Compare the lengths.



Yes—the sum of each pair of lengths is greater
than the third length.
GEOMETRY
4-6 Triangle Inequalities
TEACH! Example 5
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
GEOMETRY
4-6 Triangle Inequalities
TEACH! Example 5 Continued
Step 2 Compare the lengths.



Yes—the sum of each pair of lengths is greater
than the third length.
GEOMETRY
4-6 Triangle Inequalities
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.
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.
GEOMETRY
4-6 Triangle Inequalities
Check It Out! 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.
GEOMETRY
4-6 Triangle Inequalities
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 + 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.
GEOMETRY
4-6 Triangle Inequalities
Check It Out! 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.
GEOMETRY
4-6 Triangle Inequalities
Lesson-4 Quiz: Part I
Use the diagram for Items 1–3. Find each
measure.
1. ED
10
2. AB
14
3. mBFE 44°
GEOMETRY
4-6 Triangle Inequalities
Lesson-4 Quiz: Part II
4. Find the value of n.
16
5. ∆XYZ is the midsegment triangle of ∆WUV.
What is the perimeter of ∆XYZ?
11.5
GEOMETRY
4-6 Triangle Inequalities
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.
GEOMETRY
4-6 Triangle Inequalities
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.
GEOMETRY