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2. Inequalities in
Triangles

3. 1.4 in., 5 in., and the
8 in
2.2 in, 3 in and 5in
3.3 in, 4 in, and 8 in
4.2 in, 4 in, and 5 in
5.4 in, 5 in and 7 in

4. FToHEr LtLh Threeoere ml i(nThee Lseegments to
be the sides of a triangle,
there must be a specific
relationship among their
lengths.

5. Triangle Inequality
Theorem
In a triangle, the sum of the
lengths of any two sides
must be greater than the
length of the third side.

6. Could it be less than?
No, because if the third side was
less than the sum of the other two
sides, it would be shorter.
Therefore, it would not be long
enough to connect with the other
two sides.

7. Could it be equal to?
No, because if the third side
was equal to the sum of the
other two sides, it would be
the same length. Therefore,
it would not make a triangle.

8. Can you make a triangle in a 4in., 5in.,
and 12in.?
No.
5+4 > 9
a + b > c
b + c > a
c + a > b

9. 3 m, 4 m, and 1 m
5 yd, 13 yd, and 10 yd
24 in, 13, in, and 5 in

10. Finding the Range of the Third Side
Since the third side cannot be larger
than the other two added together, we
find the maximum value by adding the
two sides.
Since the third side and the smallest
side given cannot be larger than the
other side, we find the minimum value by
subtracting the two sides.
Difference < Third Side < Sum

11. Example: a triangle has side lengths of 6 and 12;
what are the possible lengths of the third side?
B
A
C
6 12
X = ?
12 + 6 = 18
12 – 6 = 6
Therefore:
6 < X < 18

12. Can the following lengths be the sides
of a triangle?
1. 3, 4, 9
2. 2, 8, 6
3. 5, 12, 10
Find the range for the 3rd side of the
following triangles when given the
length of two sides.
4. 5, 12
5. 16, 22
6. 10, 3

13. The Triangle Inequality Theorem
1. When you add the 2 smallest sides of a triangle, the
answer is always ___________
than the third side.
For 2-4, two sides of a triangle are given which can
be the measure of the 3rd side?
2. 10, 18
a. 8 c. 20
b. 28 d. 5
3. 20, 5
a. 23 c. 30
b. 10 d. 15
4. 2, 8
a. 10 c. 6
b. 7 d. 15