Chapter 5.1, Problem 4CE

To determine

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

Answer to Problem 4CE

  $x=56°$

  $y=39°$

  $z=85°$

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 $\overline{AD}\parallel \overline{BC}$ and $\overline{BD}$ is a transversal, so by Theorem 3-2,

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

That is,

  $m\angle z=m\angle DBC=85$

Also since $ABCD$ is a parallelogram, so by theorem 5-2,

$\angle x\cong \angle DAB$   (Opposite angles of parallelogram $ABCD$ )

That is,

  $m\angle x=m\angle DAB=56$

Now consider the triangle $△DAB$ , so by theorem 3-11,

  $m\angle ADB+m\angle DBA+m\angle BAD=180$

  $\begin{array}{c}85+y+56=180\\ y+141=180\\ y=180-141\\ y=39\end{array}$