Chapter 7, Problem 6SAP

To determine

To find: correct option representing the sum $a+b$ in terms of $x$ .

Explanation of Solution

Given information:

The figure of rectangle $PQRS$ are given below.

  

The options are as follows:

1. $90+x$
2. $190-x$
3. $180+x$
4. $270-x$
5. $360-x$

Calculation:

The angle S can be represented as follows:

  

Each angle of given rectangle (in degree) is $90°$ i.e. $\angle P=\angle Q=\angle R=\angle S=90°$ .

First evaluate angle $y°\text{ and }z°$ as follows:

  $\begin{array}{c}y°+a°+90°=180°\left[\text{Angle sum property of triangle}\right]\\ y°=180°-90°-a°\\ =90°-a°\\ \\ z°+b°+90°=180°\left[\text{Angle sum property of triangle}\right]\\ z°=180°-90°-b°\\ =90°-b°\end{array}$

Use angle sum property of rectangle to evaluate $a+b$ as shown below.

  $\begin{array}{c}\angle P+\angle Q+\angle R+\angle S=360°\\ 90°+90°+90°+\left(y°+x°+z°\right)=360°\left[\because \angle S=y°+x°+z°\right]\\ 90°+90°+90°+\left(90°-a°\right)+x°+\left(90°-b°\right)=360°\left[\text{Put values of }y°\text{ and }z°\right]\\ 450°-a°-b°+x°=360°\\ 450°-\left(a°+b°\right)+x°=360°\\ 450°-360°+x°=a°+b°\\ 90°+x°=a°+b°\\ or\\ a+b=90+x\end{array}$

Since, the sum is $90+x$ . So, option A. is correct.