Find the values of x and y that make the red lines parallel and the blue lines parallel.

The theorem 3-6 can be used which states that the relation between two lines is parallel only when a separate line travels through these two lines and the angles which are positioned on the same side are supplementary which means that their measurement will be equal to if they are added.

Name the corners of the figure.

The blue lines $\overline{AD}$and $\overline{BC}$are having a separate line $\overline{AB}$that is travelling through them and this result in formation of angles $\angle DAB$and $\angle ABC$which are positioned on same side of interior.

The red lines $\overline{AB}$and $\overline{CD}$are having a separate line $\overline{AD}$that is travelling through them and this result in formation of angles $\angle DAB$and $\angle CDA$which are positioned on same side of interior.

$x=90$According to the theorem the lines $\overline{AD}$and $\overline{BC}$are having a parallel relation and the angles $\angle DAB$and $\angle ABC$represents the supplementary angles. Determine the value of x by using the property of $\angle DAB$and $\angle ABC$.

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

$\left(x-40\right)+\left(x+40\right)=180$

$x-40+x+40=180$

$x+x-40+40=180$

$2x=180$

$x=\frac{180}{2}$

$x=90$

Substitute the value of x into the expression that represents $\angle DAB$to determine its actual value.

$$\begin{gathered} m\angle DAB=x-40 \\ =90-40 \\ =50 \end{gathered}$$

According to the observation and theorem the lines $\overline{AB}$and $\overline{CD}$are having a parallel relation and the angles $\angle DAB$and $\angle CDA$ represents the supplementary angles. Determine the value of y by using the property of $\angle DAB$and $\angle CDA$.

$\angle DAB+m\angle CDA=180$

$50+y=180$

$y=180-50$

$y=130$