91Ó°ÊÓ

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

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

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

$∠DAB+m∠ABC=180$

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

$x−40+x+40=180$

$x+x−40+40=180$

$2x=180$

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

$x=90$

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

$$\begin{gathered} m∠DAB=x−40 \\ =90−40 \\ =50 \end{gathered}$$

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

$∠DAB+m∠CDA=180$

$50+y=180$

$y=180−50$

$y=130$