Triangle Inequalities
TRIANGLE INEQUALITIES
In this lesson, we will investigate two very basic geometric inequalities. While there are other inequality
relationships in triangles, these two are the ones most commonly used.
LICORICE
INVESTIGATION:
MATERIALS NEEDED:
This first investigation involves the lengths of the sides of a triangle.
6 Licorice
Pieces*/ Student
Triangle
Inequality WS
1. Cut or tear your licorice pieces into various lengths (whole numbers work more easily than
fractional pieces).
2. Select three pieces of licorice (rank order them from smallest to largest based upon their
lengths).
3. Use three pieces of licorice to construct a triangle, if possible.
4. Collect your data on the chart provided.
5. Consider your charts to be completed when your group has collected data for 3 examples
where the sides formed triangles and for 3 examples where the sides did NOT form
triangles.
6. Using your collected data, complete the following:
TRIANGLE INEQUALITY THEOREM: The sum of the lengths of any two sides of a triangle
is ________________ the length of the third side.
* You may want to use spaghetti instead for this investigation.
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Triangle Inequalities
Lengths of Sides
1. s = 3inches (SAMPLE)
m = 4 inches
l = 5 inches
2. s = _____
m = _____
l = _____
3. s = _____
m = _____
l = _____
4. s = _____
m = _____
l = _____
5. s = _____
m = _____
l = _____
6. s = _____
m = _____
l = _____
∆ Formed? (Circle One)
YES OR NO?
YES OR NO?
YES OR NO?
YES OR NO?
YES OR NO?
YES OR NO?
Comparison of Sides (<, >, or =)
s + m _____ l 3 + 4 > 5
m + l _____ s 4 + 5 > 3
l + s _____ m 5 + 3 > 4
s + m _____ l
m + l _____ s
l + s _____ m
s + m _____ l
m + l _____ s
l + s _____ m
s + m _____ l
m + l _____ s
l + s _____ m
s + m _____ l
m + l _____ s
l + s _____ m
s + m _____ l
m + l _____ s
l + s _____ m
TRIANGLE INEQUALITY THEOREM: The sum of the lengths of any two sides of a triangle
is ________________ the length of the third side.
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Triangle Inequalities
INVESTIGATION: The second relationship involves the lengths of the sides of a triangle in relation to the
triangle's angles.
MATERIALS NEEDED:
On a clean sheet of paper, construct a large acute triangle on the top half of your paper, and a large obtuse
triangle in the bottom half. For each triangle:
Step 1: Measure the angles in the triangle. Label the largest,  L, the second largest,  M, and the smallest, 
S.
Step 2: Measure the three sides. Which is the largest? Label it by placing a lower case l near the middle of the
side. Which side is the second largest? Label it m in the same way. Which side is the smallest? Label it s.
Which side, s, m, or l is opposite the largest angle? Which side is opposite the second largest angle? Which side
is opposite the smallest angle?
Use your results to complete the following: In a triangle, ____________________________
_______________________________________________________________________.
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Triangle Inequalities
EXERCISE SET A
For each set of lengths, determine whether it is possible to draw a triangle with sides of the given measures. If
possible, write yes. If not possible, write no. Be prepared to explain your reasoning.
1. 3cm, 4 cm, 5cm
4. 3.5 cm, 4.5 cm, 7 cm
2. 4 m, 5m, 9m
5. 4”, 5”, 7 ½ “
3. 5 ft., 6ft., 12 ft.
6. .5m, .6m, 12 cm
EXERCISE SET B
In problems 1 and 2, the letter on each side of the ∆s indicates the size of that size. In problems 3 and 4, the
letter indicates the size of the angle. Rank order the letters from large to small. NOTE: You may need to use ∆
Sum to determine missing angle measures in some triangles.
TEXT: Page 283: 31-42 all.
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