This document discusses triangle inequalities and properties related to triangle side lengths and angle measures. It provides examples of: 1) Determining the order of angles from smallest to largest given side lengths 2) Determining the order of side lengths from smallest to largest given angle measures 3) Applying the triangle inequality theorem which states the sum of any two side lengths must be greater than the third side length.

1. Triangle Inequalities
The student is able to (I can):
• Analyze the relationship between the angles of a triangle
and the lengths of the sidesand 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.
AT > AC → m∠C > m∠T
m∠C > m∠T → AT > AC
A
C
T
m∠C > m∠T → AT > AC

3. Example: Given the side lengths, put the
angles in order from smallest to
largest.
A
19 16
∠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 N
31

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

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, 124, 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?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 < 308 < x < 30