Question

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

Short answer

The required value of xis 35 and the value of y is 20.

Step-by-step solution

Step 1. Determine the theorem or postulate

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.

Step 2. Label the provided figure and observe it

Name the corners of the figure.

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

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

Step 3. Use the theorem 3-5 and determine the value of x

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

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

Step 4. Use the theorem 3-6 and determine the value of x

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

$m\angle BCE+m\angle CED=180$

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

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

$m\angle BCE=m\angle BCD+m\angle DCE$

$=105+2y$

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

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