5.6: Inequalities in 2 Triangles

Use the Hinge Theorem & its converse to find a range of values for a given side or angle.

Think About It! The angle that a person makes as he or she is sitting changes with the task. The diagram shows

the position of a student as his desk. In which position is the angle measure at which he is sitting the greatest?

The least? Justify your answer.

Theorem Hypothesis Conclusion Hinge Theorem:

If 2 sides of one triangle are congruent to 2 sides of another triangle and the included

angles are not congruent, then the longer 3rd side is across from the _________________

included angle.

Converse of the Hinge Theorem

If 2 sides of one triangle are congruent to 2 sides of another triangle and the third sides

are not congruent, then the larger 3rd angle is across from the _________________________

included side.

Directions: Write an inequality to compare the given measures

Example 1: SR and PQ Example 2: mG and mL

Directions: Find a range of values for x

Example 3: Example 4:

Example 5: Example 6: Challenge!

Example 7: Two cyclists start from the same location and travel in opposite directions for 2 miles each. Then the

first cyclist turns right 90° and continues for another mile. At the same time, the second cyclist turns 45° left and

continues for another mile.

a) At this point, which cyclist is closer to the original starting point? Draw a picture to support your answer.

b) What is the range in distance from the starting point to where cyclist 1 currently is? Cyclist 2? Are they same

or different? Explain.