Chapter 5.1, Problem 27WE

To determine

To fill: the blanks with the values of x and $\angle CED$ for a given parallelogram.

Answer to Problem 27WE

The value of x is $10$

The value of $\angle CED$is $70°$

Explanation of Solution

Given information:

A diagram representing a parallelogram with the measure of angles

Formula used:

1) The opposite angles of a parallelogram are in congruence which implies they have the same measure.

2) The alternate interior angles of the parallelogram are equal.

Calculation:

Consider the figure given below.

  

Based on the property in (2), the angles are in the relation,

  $\angle 1=\angle 3$......(A)

  $\angle 2=\angle DEK$......(B)

Now, substituting the values of angles in the above equations, it gets modified as,

  $\begin{array}{l}3x=x^2-70\\ x^2-3x-70=0\end{array}$

Solving the above quadratic equation gives the roots as,

  $x=10,-7$

Since only positive values are considered, the acceptable value of $x=10$ .

Now, $\angle CED$ is expressed as,

  $\angle CED=\angle DEK+x^2-70$

Using (B), the above equation can be rewritten as,

  $\angle CED=4x+x^2-70$

Substituting the value of x in the above equation leads to,

  $\begin{array}{c}\angle CED=4\left(10\right)+{10}^2-70\\ =40+100-70\\ =70°\end{array}$