Question

Find the values $x$of $y$and

Short answer

Values are $\mathit{x}\mathbf{=}\mathbf{30}$and $\mathit{y}\mathbf{=}\mathbf{5}$

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 $\overline{BD}$is transversal to the parallel lines $\overline{AB},\overline{CD}$

Line segment $\overline{DF}$is perpendicular to the parallel lines$\overline{CD},\overline{EF}$

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

$\angle ABD$and $\angle BDC$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.

$m\angle ABC+m\angle BDC=180$

$70+4x-2y=180$

$4x-2y=180-70$

$4x-2y=110\left(1\right)$

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

$\angle CDF$and $\angle DFE$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.

$m\angle CDF+m\angle DFE=180$

$4x+2y+50=180$

$4x+2y=180-50$

$4x+2y=130\left(2\right)$

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

$$\begin{gathered} 4x-2y=110 \\ \underline{4x-2y=130} \\ 8x=110 \\ x=30 \end{gathered}$$

Step 6. Substitute 30 for x in equation (1)

$$\begin{gathered} 4\left(30\right)-2y=110 \\ -2y=110-120 \\ y=\frac{-10}{-2} \\ y=5 \end{gathered}$$

Thus, $x=30$and $y=5$