91Ó°ÊÓ

The theorem 3-5 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 alternate interior side represents the congruent angles.

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 $180°$if they are added.

Name the corners of the figure.

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

The red lines $\stackrel{¯}{BC}$and $\stackrel{¯}{DE}$are having a separate line $\stackrel{¯}{CE}$that is travelling through them and this result in formation of angles $∠BCE$and $∠CED$which are positioned on same side of interior.

According to the theorem the lines $\stackrel{¯}{AB}$and $\stackrel{¯}{CD}$are having a parallel relation and the angles $∠BCD$and $∠ABC$ represents the congruent angles. Determine the value of x by using the property of $∠BCD$and $∠ABC$.

$$\begin{gathered} m∠ABC=m∠BCD \\ 3x=105 \\ x=\frac{105}{3} \\ x=35 \end{gathered}$$

According to the observation and theorem the lines $\stackrel{¯}{AB}$and $\stackrel{¯}{CD}$are having a parallel relation and the angles $∠BCE$and $∠CED$represents the supplementary angles. Make an equation by using the property of $∠DAB$and $∠CDA$.

$m∠BCE+m∠CED=180$

According to the figure $∠CED$is same as x which means the value of $∠CED$will be 35.

$∠BCE$represents the addition of two angles which are $∠BCD$and $∠DCE$so:

$m∠BCE=m∠BCD+m∠DCE$

$=105+2y$

Using the values of $∠BCE$and $∠CED$calculate the value of y.

$$\begin{gathered} m∠BCE+m∠CED=180 \\ 105+2y+35=180 \\ 105+35+2y=180 \\ 140+2y=180 \\ 140+2y=180 \\ 2y=180−140 \\ 2y=40 \\ y=\frac{40}{2} \\ y=20 \end{gathered}$$