Chapter 5.1, Problem 22WE

To determine

To find: the values of x and y for a given parallelogram.

Answer to Problem 22WE

The value of x is $65$

The value of y is $20$

Explanation of Solution

Given information:

A diagram representing a parallelogram with its angles being given.

Formula used:

1) Property of alternate interior angles being equal when two parallel lines cut by transversal.

2) Sum of Angles on a straight line is 180⁰

Calculation:

Consider the figure given below.

  

From the figure, it can be observed that,

  $\left(x-y\right)+\left(x+y\right)+\angle AOD=180°$

  $2x+\angle AOD=180°$ ……… (1)

Now, since the diagonals bisect each other, the sides can be related as,

  $\begin{array}{l}AO=OC\\ OD=OB\end{array}$

Also,

  $\begin{array}{c}\angle AOD=180°-130°\\ =50°\end{array}$

Substituting the value of $\angle AOD$ in (1),

  $\begin{array}{l}2x+50=180°\\ 2x=130°\\ x=65\end{array}$

Now, using the property of alternate interior angles,

  $\left(x+y\right)=85°$

Substituting the value of x in the above equation, the value of y can be calculated as,

  $\begin{array}{c}\left(65+y\right)°=85°\\ y=85-65\\ =20\end{array}$