Chapter 5.2, Problem 23PSB

To determine

To Find: Whether $x=25,\text{if }\overleftrightarrow{PQ}\cancel{\parallel }\overleftrightarrow{RS}$ , with explanation.

Answer to Problem 23PSB

  $\text{No},x\ne 25,\text{if }\overleftrightarrow{PQ}\cancel{\parallel }\overleftrightarrow{RS}$

Explanation of Solution

Given:

  

Concept Used:

When two parallel lines is intersected by a transversal corresponding angles are equal.

Calculation:

Here, we have

  $\begin{array}{l}m\angle QBC=\left(2x\right)°\text{ and}m\angle RCD=\left(4x+30\right)°\\ \Rightarrow m\angle PBC=\left(180-2x\right)°\\ \text{Now, if }\overleftrightarrow{PQ}\parallel \overleftrightarrow{RS}\\ m\angle PBC=m\angle RCD\\ \Rightarrow \left(180-2x\right)°=\left(4x+30\right)°\\ \Rightarrow -\left(6x\right)°=-150°\\ \Rightarrow x=25,\text{but }\overleftrightarrow{PQ}\cancel{\parallel }\overleftrightarrow{RS},\text{which is not possible}\text{.}\\ \Rightarrow x\ne 25,\text{if }\overleftrightarrow{PQ}\cancel{\parallel }\overleftrightarrow{RS}\end{array}$