The hinge theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. It can be used to determine if one angle is larger than another based on the lengths of the opposite sides. The document provides examples of using the hinge theorem to compare angles and their opposite sides in different triangles and determine inequalities between variables.

1. Hinge Theorem
The student is able to (I can):
• Analyze the relationship between angles and the sides
opposite them
• Use the Hinge Theorem to solve problems

2. 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
FE
D
∠∠ > ⇔ >Em mB AC DF

3. Example Based on the diagram below, which car is
farther away from the mall?

4. Example Based on the diagram below, which car is
farther away from the mall?
Since 100° is greater than 90°, car B is
farther away from the mall.

5. Example Based on the diagram, which angle is
bigger, ∠1 or ∠2?

6. Example Based on the diagram, which angle is
bigger, ∠1 or ∠2?
Since 36 > 34, then m∠1 > m∠2.

7. Example: Find the range of values for x.
1.
2.
(2x+8)°
26°
7
8
64°
35°
x+7 2x−5

8. 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