Exploring Triangle Inequalities
Exploring Triangle Inequalities
Exploring Triangle Inequalities
Resource
Locker
A B
D Measure and record the lengths of the three sides of your triangle.
C C
A B A B
Reflect
1. Use the side lengths from your table to make the following comparisons. What do
you notice?
AB + BC ? AC BC + AC ? AB AC + AB ? BC
2. Measure the angles of some triangles with a protractor. Where is the smallest angle
in relation to the shortest side? Where is the largest angle in relation to
the longest side?
3. Discussion How does your answer to the previous question relate to isosceles
triangles or equilateral triangles?
A The sum of any two side lengths of a triangle is greater than the third side length.
A AB + BC > AC
AB + BC > AC
BC + AC > AB
BC + AC > AB
B C AC + AB > BC
B C AC + AB > BC
To be able to form a triangle, each of the three inequalities must be true. So, given three side
lengths, you can test to determine if they can be used as segments to form a triangle. To show
that three lengths cannot be the side lengths of a triangle, you only need to show that one
of the three triangle inequalities is false.
A 4, 8, 10
? ? ?
4 + 8 > 10 4 + 10 > 8 8 + 10 > 4
12 > 10 ✓ 14 > 8 ✓ 18 > 4 ✓
Conclusion: The sum of each pair of side lengths is greater than the third length.
So, a triangle can have side lengths of 4, 8, and 10.
B 7, 9, 18
? ? ?
+ > 18 + > 9 9 + >
Conclusion:
Reflect
5. How do you know that the Triangle Inequality Theorem applies to all equilateral triangles?
Your Turn
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Determine if a triangle can be formed with the given side lengths. Explain your
reasoning.
Example 2 Find the range of values for x using the Triangle Inequality Theorem.
A Find possible values for the length of the third side using the Triangle Inequality Theorem.
12
x
10
x + 10 > 12 x + 12 > 10 10 + 12 > x
x> 2 x > -2 22 > x
2 < x < 22
Ignore the inequality with a negative value, since a triangle cannot have a negative side
length. Combine the other two inequalities to find the possible values for x.
B 15
15
<x<
8. Discussion Suppose you know that the length of the base of an isosceles triangle is 10,
but you do not know the lengths of its legs. How could you use the Triangle Inequality
Theorem to find the range of possible lengths for each leg? Explain.
Your Turn
Find the range of values for x using the Triangle Inequality Theorem.
9. 10.
21 14 x 9
x
18
C C C C
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A A B B A A B B
As side AC gets
As side AC gets
shorter, m∠Bmapproaches
shorter, ∠B approaches
0° 0° As side AC gets
As side AC gets
longer, m∠Bmapproaches
longer, ∠B approaches
180°180°
AC > BC
m∠B > m∠A
A B
C C
A B
21 22
15 12 A B
20
A B
10 Longest side length:
Longest side length: AC
Greatest angle measure:
Greatest angle measure: m∠B
Shortest side length:
Shortest side length: AB
Least angle measure:
Least angle measure: m∠C
Order of angle measures from
Order of angle measures from least to greatest:
least to greatest:
m∠C, m∠A, m∠B
Your Turn
For each triangle, order its angle measures from least to greatest.
11. 40 A 12. C
B
25
7
15
32
A B
24
C
A A B B A A B B
As m∠B approaches
As m∠B 0°, side
approaches AC gets
0°, side shorter
AC gets shorter As m∠B approaches
As m∠B 180°,180°,
approaches AC gets
sideside longer
AC gets longer
C C
A B
50° 45°
A 30° 100° B
A 70° 65° B
Greatest angle measure: m∠B
Greatest angle measure:
Longest side length: AC
Longest side length:
Least angle measure: m∠A
Least angle measure:
Shortest side length: BC
Shortest side length:
Order of side lengths from least to greatest: BC,
AB, AC Order of side lengths from least
to great:
Your Turn
For each triangle, order the side lengths from least to greatest.
13. A 14. C
5°
15°
160°
C B 60° 30°
A B
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Elaborate
15. When two sides of a triangle are congruent, what can you conclude about the angles opposite those sides?
16. What can you conclude about the side opposite the obtuse angle in an obtuse triangle?
17. Essential Question Check-In Suppose you are given three values that could represent the side lengths
of a triangle. How can you use one inequality to determine if the triangle exists?
Determine whether a triangle can be formed with the given side lengths.
5. 10 ft, 3 ft, 15 ft 6. 12 in., 4 in., 15 in.
Find the range of possible values for x using the Triangle Inequality Theorem.
9. 10.
x 3 x
5
8
12
12. Analyze Relationships Suppose a triangle has side lengths AB, BC, and x, where AB = 2 · BC. Find the
possible range for x in terms of BC.
13 6 3.7
3.2
C
F
15. Analyze Relationships Suppose a triangle has side lengths PQ, QR, and PR, where PR = 2PQ = 3QR.
Write the angle measures in order from least to greatest.
For each triangle, write the side lengths in order from least to greatest.
16. A 17. E
65° 79°
33° F
B 45°
70° D
C
18. In △ JKL, m∠J = 53°, m∠K = 68°, and 19. In △ PQR, m∠P = 102° and m∠Q = 25°.
m∠L = 59°.
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23. Represent Real-World Problems Three cell phone towers form a triangle,
△ PQR. The measure of ∠Q is 10° less than the measure of ∠P. The measure of ∠R
is 5° greater than the measure of ∠Q. Which two towers are closest together?
C C
A B A B
B
26. Given the information in the diagram, prove that m∠DEA < m∠ABC.
9
D
4 7
A 9 E 2 C
27. An isosceles triangle has legs with length 11 units. Which of the following could be
the perimeter of the triangle? Choose all that apply. Explain your reasoning.
a. 22 units
b. 24 units
c. 34 units
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d. 43 units
e. 44 units
D G E
B
G
48° E 94° H
A
53°
F
C D 58°
I