5.5 Hinge Theorem

​Hinge Theorem

By Rachel Courtney

Multiple Choice

SAS congruence is how we prove triangles congruent. This happens when...

• two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle.

• two sides and an angle of one triangle are congruent to two sides and an angle of another triangle.

• two angles and the included side of one triangle are congruent to two angles and the included side of another triangle.

• two angles and a side of one triangle are congruent to two angles and a side of another triangle.

Multiple Choice

• A

• B

• C

• D

What happens if...

the angles are not congruent but the two sides are?

Multiple Choice

Two sides of each triangle in the circle are formed

from the radii of the circle. Compare EF and FG.

• EF = FG

• EF < FG

• EF > FG

• Not enough information is given.

Multiple Choice

BC _____ EF

• <

• >

• =

• No conclusion can be made.

Multiple Choice

Compare QT and ST

• $$QT\ <\ ST$$

• $$QT\ >\ ST$$

• $$QT\ =\ ST$$

• $$\overline{QT}\ \cong\ \overline{ST}$$

Multiple Choice

Compare RT and VX

• RT=VX

• RT<VX

• RT>VX

• RT $$\cong$$ VX 

THE HINGE THEOREM

If two sides of a triangle are congruent to two sides of another triangle and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.

CONVERSE OF THE HINGE THEOREM

If two sides of a triangle are congruent to two sides of another triangle and the third side of the first is longer than the third side of the second, then the included angle in the first triangle is greater than the included angle in the second triangle.

Multiple Choice

What are the possible values for x?

• x < 16

• 4 < x < 16

• x < 40

• 4 < x < 40

Multiple Choice

Find the range of possible values for x.

• $$2\ <x<6$$

• $$6<x<15\ \frac{1}{3}$$

• $$6<x<22$$

• $$2<x<22$$

Multiple Choice

Range of Possible Values for: x

• 8>x>26

• 7<x<25

• 8<x<26

• 7>x>25

Multiple Choice

Find the range of possible values for x

• $$x>7$$

• $$-\frac{17}{3}<x>7$$

• $$-\frac{17}{3}<x<7$$

• $$x<7$$

Multiple Choice

Find the range of possible values of x.

• $$-20<x<21$$

• $$8<x<37$$

• $$34<x<37$$

• $$8<x<24$$