Chapter 15.3, Problem 9PSB

To determine

To Prove: The side of the given triangleQR < PS.

Explanation of Solution

Given:

  $\overline{PQ}=\overline{PR}=\overline{RS}$

  

Concept:

The Hinge Theorem: It states that if two sides of two triangles are congruent and angle of one is greater than the angle of second then the third side of the triangle will be greater than the another.

Calculation:

  $\begin{array}{l}\to PQ=PR\left(\text{Given}\right)\\ \to PR=RS\left(\text{Given}\right)\\ \text{⇒}\angle QPR<\angle PRS\left(\text{Exterior }\angle \text{ of a triangle is greater than its remote interior }\angle \right)\\ \text{⇒}QR<PS\left(\text{SAS inequality or Hinge theorem}\right)\end{array}$

Conclusion:

Thus, by using the SAS inequality theorem it is proved that $QR<PS$ .