Chapter 3.2, Problem 15WE

To determine

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

Answer to Problem 15WE

  $x=70$

  $y=12$

  $z=38$

Explanation of Solution

Given:

The diagram:

Concept Used:

Postulate 10: If two parallel lines are cut by a transversal then the corresponding angles are congruent

Theorem 3-3: If two parallel lines are cut by a transversal then the same-side interior angles are supplementary.

Label the given figure as shown below:

In the above figure, lines $\overline{AB}\parallel \overline{CD}$ and $\overline{XY}$ is a transversal, so by theorem 3-3,

  $m\angle BPQ+m\angle PQD=180$ (Same-side interior angles)

This implies that,

  $\begin{array}{c}110+x=180\\ x=180-110\\ x=70\end{array}$

Now lines $\overline{XY}\parallel \overline{UV}$ and $\overline{CD}$ is a transversal, so by postulate 10,

  $\angle PQR\cong \angle SRD$ (Corresponding angles)

That is,

  $\begin{array}{c}m\angle SRD=m\angle PQR\\ 5y+10=70\\ 5y=60\\ y=12\end{array}$

Now lines $\overline{AB}\parallel \overline{CD}$ and $\overline{UV}$ is a transversal, so by postulate 10,

  $\angle USR\cong \angle SRD$ (Corresponding angles)

That is,

  $\begin{array}{c}m\angle USR=m\angle SRD\\ z+32=70\\ z=70-32\\ z=38\end{array}$