Question

Find the values of $x$and $y$

Short answer

Values are $\mathit{x}\mathbf{=}\mathbf{25}$and $\mathit{y}\mathbf{=}\mathbf{10}$

Step-by-step solution

Step 1. Label the given figure

Step 2. Observations from figure

Arrowheads denotes line segment $\overline{AB}$, $\overline{CD}$, and $\overline{EF}$are parallel

Line segment $\overline{AC}$is transversal to the parallel lines $\overline{AB},\overline{CD}$,

Line segment $\overline{CE}$is transversal to the parallel lines $\overline{CD}$,$\overline{EF}$

Step 3. Find first relation between x degree and y degree

$\angle DCA$and $\angle CAB$forms a pair of same-side interior angles

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

$m\angle DCA+m\angle CAB=180$

$2x+y+120=180$

$2x+y=180-120$

$2x+y=60\left(1\right)$

Step 4. Find second relation between x degree and y degree

$\angle DCE$and $\angle CEF$forms a pair of same-side interior angles

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

role="math" localid="1637760125240" $m\angle DCE+m\angle CEF=180$

$2x-y+140=180$

$2x-y=180-140$

$2x-y=40\left(2\right)$

Step 5. Add equations (1) and (2)

$$\begin{gathered} 2x+y=60 \\ \underline{2x-y=40} \\ 4x=100 \\ x=25 \end{gathered}$$

Step 6. Substitute 25 for x in equation (1)

$2\left(25\right)+y=60$

$50+y=60$

$y=60-50$

$y=10$

Thus, $x=25$and$y=10$