Chapter 5.6, Problem 23PPS

To determine

Toprove:The side $RS$ is greater than $QR$ .

Explanation of Solution

Given information:

The given figure is

  

  $\overline{LK}\cong \overline{JK},\overline{RL}\cong \overline{RJ}$

  $K$ is the midpoint of $\overline{QS}$

  $m\angle SKL>m\angle QKJ$

Formula Used:

Hinge theorem is

If two sides of a triangle correspond to two sides of another triangle, and the angle of inclination of the first triangle is greater than the angle of inclination of the second triangle, so the sides of the first triangle are longer than the sides of the third side of the second triangle.

Prove:

Given that $K$ is the midpoint of $\overline{QS}$ .

Thus, $\overline{QK}\cong \overline{SK}$

Since, $m\angle SKL>m\angle QKJ$ . Thus, by Hinge theorem.

  $SL>QJ$

Add the segment $RL,RJ$ in the respective sides.

  $SL+RL>QJ+RJ$

Since, $RS=SL+RL$ and $QR=QJ+RJ$ .

Thus, replace the values, we get,

  $RS>QR$

Hence, proved.