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

1. Obj. 22 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. Thm 5-5-1
Thm 5-5-2
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 ®mÐC > mÐT
mÐC > mÐT ® AT > AC

3. 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

4. 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.

5. 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

6. Hinge Theorem
and Converse
If two sides of one triangle are congruent
to two sides of another triangle …
• If the included angles are not congruent,
then the longer third side is across from
the larger included angle.
• If the third sides are not congruent,
then the larger included angle is across
from the longer third side.
A
B C
D
E F
mÐB mÐEÛAC DF

7. Example: Find the range of values for x.
1.
2.
(2x+8)°
26°
7
8
+
0 2x 8 26
− 8 2x
18
− 4 x
9
64°
35°
x+7 2x−5
0 2x − 5 and 2x − 5 x + 7
5 2x
2.5
x
−
x 5 7
x 12
2.5 x 12