Chapter 9.5, Problem 20WE

To determine

To calculate: The unknown angles.

Answer to Problem 20WE

  $x=9°$ and $m\angle D=101°$

Explanation of Solution

Given information: $m\angle A=x^2,m\angle B=9x-2$ and $m\angle C=11x$

Formula used: We know the sum of the opposite angles of a quadrilateral inscribed in a circle is $180°$ . We can write $m\angle A+m\angle C=180°$ and $m\angle B+m\angle D=180°$

Calculation: Using the above relation.

  

Plugging the given value we get,

  $\begin{array}{l}m\angle A+m\angle C=180°\\ x^2+11x=180\\ x^2+11x-180=0\end{array}$

Now, factoring.

  $\begin{array}{l}{x}^2+11x-180=0\\ x^2-9x+20x-180=0\\ x(x-9)+20(x-9)=0\\ (x+20)(x-9)=0\end{array}$

We get $x=9,x=-20$ Ignoring negative values of the angle.

And

  $\begin{array}{l}m\angle B+m\angle D=180°\\ 9x-2+m\angle D=180\\ 9\times 9-2+m\angle D=180\\ m\angle D=180-79\\ m\angle D=101°\end{array}$