Chapter 3.2, Problem 18WE

To determine

To find: the values of x and y .

Answer to Problem 18WE

  $x=25$

  $y=10$

Explanation of Solution

Given:

The diagram:

Concept Used:

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:

Now, lines $\overline{AB}\parallel \overline{CD}$ and $\overline{AC}$ is a transversal, so by theorem 3-3,

  $m\angle BAC+m\angle ACD=180$ (Supplementary angles)

  $\begin{array}{c}120+2x+y=180\\ 2x+y=180-120\\ 2x+y=60\end{array}$

Similarly, lines $\overline{CD}\parallel \overline{EF}$ and $\overline{CE}$ is a transversal, so by theorem 3-3,

  $m\angle DCE+m\angle CEF=180$ (Supplementary angles)

  $\begin{array}{c}2x-y+140=180\\ 2x-y=180-140\\ 2x-y=40\end{array}$

Now, consider the equations:

  $2x+y=60$ ….. (1)

  $2x-y=40$ ….. (2)

Add equations (1) and (2),

  $\begin{array}{c}\left(2x+y\right)+\left(2x-y\right)=60+40\\ 2x+y+2x-y=100\\ 2x+2x+y-y=100\end{array}$

  $\begin{array}{c}4x=100\\ x=\frac{100}{4}\\ x=25\end{array}$

Substitute $x=25$ in equation (1),

  $\begin{array}{c}2\left(25\right)+y=60\\ 50+y=60\\ y=60-50\\ y=10\end{array}$