The hinge theorem states that if two sides of one triangle are congruent to two sides of another triangle, then the included angles are either congruent or the angles are not congruent and the longer side is opposite the larger angle. The document provides examples of using the hinge theorem to determine which angle is larger based on opposite side lengths and which object is farther from a point based on included angle measures.

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 andHinge Theorem andHinge Theorem andHinge Theorem and ConverseConverseConverseConverse – 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 2 8 26
8 2 18
4 9
x
x
x
< + <
− < <
− < <
64°
35°
x+7 2x−5
0 2 5 and 2 5 7x x x< − − < +
5 2
2.5
x
x
<
<
5 7
12
x
x
− <
<
2.5 12x< <