Example 1:

If  is the median of  and , then it follows from the Hinge Theorem that , and therefore :

Example 2: 

If  is equilateral, , and , it follows from the Hinge Theorem that sides  and  are shorter than the equal sides ,  and .  This is because  and  so each measures less than 60o.  Therefore  is the largest angle in , which makes side  the largest side in that triangle.

Example 3:

Jan and Jon are hikers.  They start at the visitor center and walk in opposite directions for 2 miles.  Then Jan turns to her right at an angle of 20^o, and Jon turns to his right at an angle of 30^o.  They each continue hiking for another mile and a half when both stop to rest.  Who is farther (as the crow flies) from the visitor center?

Solution:

Draw the segments from each person's final position to the visitor center, and find the supplementary angles formed by their turning angles.  Then you have two triangles, and , with  and :

Since , it follows from the Hinge Theorem that , so Jan is farther from the visitor center.