Chapter 5.1, Problem 5CE

To determine

To find: the values of x, y and z.

Answer to Problem 5CE

  $x=110°$

  $y=40°$

  $z=70°$

Explanation of Solution

Given:

The diagram:

Consider the figure shown below:

Here it is given that $ABCD$ is a parallelogram.

So, its opposite sides are parallel. That is $\overline{AB}\parallel \overline{CD}$ and $\overline{AD}\parallel \overline{BC}$

Now, lines $\overline{AB}\parallel \overline{CD}$ and $\overline{BD}$ is a transversal, so by Theorem 3-2,

  $\angle z\cong \angle ABD$   (Alternate interior angles)

That is,

  $m\angle z=m\angle ABD=70$

Also $\angle AOD$ and $\angle BOC$ are vertical angles, so by theorem 2-3,

  $\angle x\cong \angle AOD$   (Vertical angles)

That is,

  $m\angle x=m\angle AOD=110$

Now for the triangle $△AOB$ , $\angle BOC$ is the exterior angle corresponding to the two remote interior angles $\angle BAO$ and $\angle ABO$ , so by theorem 3-12,

  $m\angle BAO+m\angle ABO=m\angle BOC$

  $\begin{array}{c}y+70=110\\ y=110-70\\ y=40\end{array}$