Chapter 5.2, Problem 22PSB

To determine

To Find: Find the restriction on $x$ .

Answer to Problem 22PSB

  ${81}^{\circ }<x<{143}^{\circ }$

Explanation of Solution

Given:

  

Concept Used:

Exterior angle is equal to the sum of two interior opposite angles.

  $\text{Sum of interior angles of triangle }={180}^{\circ }$

Calculation:

Since, $\text{Sum of interior angles of triangle }={180}^{\circ }$

  $\begin{array}{l}\Rightarrow \angle LKM+\angle KLM+\angle LMK={180}^{\circ }\\ \Rightarrow {81}^{\circ }+\angle KLM+{37}^{\circ }={180}^{\circ }\\ \Rightarrow \angle KLM={180}^{\circ }-{118}^{\circ }\\ \Rightarrow \angle KLM={62}^{\circ }\\ \because \angle KLM=\angle KLP+\angle MLP\\ \Rightarrow \angle KLP={62}^{\circ }-\angle MLP\left(1\right)\\ \text{Also},\angle LPM+\angle MLP+\angle LMP={180}^{\circ }\\ \Rightarrow x+\angle MLP+{37}^{\circ }={180}^{\circ }\\ \Rightarrow x={143}^{\circ }-\angle MLP\\ \text{Since, exterior angle is equal to the sum of two interior opposite angles}\text{.}\\ x={81}^{\circ }+\angle KLP\\ \Rightarrow {81}^{\circ }<x<{143}^{\circ }\end{array}$