Analyze the relationship between the angles of a triangle and the lengths of the sides Determine allowable lengths for sides of triangles

1. Triangle Inequalities
The student is able to (I can):
• Analyze the relationship between the angles of a triangle
and the lengths of the sides
• Determine allowable lengths for sides of triangles

2. If two sides of a triangle are not congruent, then the larger
angle is opposite the longer side.
If two angles of a triangle are not congruent, then the longer
side is opposite the larger angle.
A
C
T
AT > AC → m∠C > m∠T
m∠C > m∠T → AT > AC

3. Example: Given the side lengths, put the angles in order
from smallest to largest.
∠P is across from 16, ∠N is across from 19, and ∠A is
across from 31, so it would be: ∠P, ∠N, and ∠A
P
A
N
19
31
16

4. Example: Given the angle measures, put the side lengths in
order from smallest to largest.
First, we have to calculate m∠E:
m∠E = 180- (70+30) = 80°
So the sides would be:
TE < EN < TN
T
E
N
70° 30°

5. Triangle Inequality Theorem
The sum of any two side lengths of a triangle is greater than
the third side length.
Example:
1. Which set of lengths forms a triangle?
4, 5, 10 7, 9, 12
4 + 5 < 10 7 + 9 > 12

6. Note: To find a range of possible third sides given two sides,
subtract for the lower bound and add for the upper
bound.
Examples:
2. What is a possible third side for a triangle with sides 8 and
14?
14 – 8 = 6 lower bound
14 + 8 = 22 upper bound
The third side can be between 6 and 22.

7. 3. What is the range of values for the third side of a triangle
with sides 11 and 19?
19 – 11 = 8 lower bound
19 + 11 = 30 upper bound
8 < x < 30