5-6 Inequalities in Two Triangles

The Hinge Theorem

• When you close a door, the angle between the door and the frame (at the hinge) gets smaller.

Hinge Theorem: If two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle.

Using the Hinge Theorem

• What inequality relates SK to YU?

What inequality relates LN to OQ?

The Converse of the Hinge Theorem

• If two sides of one triangle are congruent to two sides of another triangle, and the third sides are not congruent, then the larger included angle is opposite the longer third side.

Compare mBAC and mDAC.

Compare EF and FG.

Compare BC and AB.

Find the range of values for k.

Step 1 Compare the side lengths in ∆MLN and ∆PLN.

By the Converse of the Hinge Theorem, mMLN > mPLN.

LN = LN LM = LP MN > PN

5k – 12 < 38

k < 10

Substitute the given values.

Add 12 to both sides and divide by 5.

Step 2 Since PLN is in a triangle, mPLN > 0°.

The range of values for k is 2.4 < k < 10.

5k – 12 > 0k < 2.4

Substitute the given values.

3. Find the range of values for z.

–3 < z < 7

Lesson Quiz: Part I 1. Compare mABC and mDEF.

2. Compare PS and QR.

mABC > mDEF

PS < QR