Question

In the exercise 3 quad.ABCD is a parallelogram. Find the values of x, y, and z.

Short answer

The value of x, y, and z is $65°,65°$and $115°$respectively.

Step-by-step solution

Step 1. Determine the value of angle x

According to the provided information, the opposite sides of the provided figure are parallel. That is $\overline{AB}\parallel \overline{CD}$and $\overline{AD}\parallel \overline{BC}$.

Now, lines$\overline{AD}\parallel \overline{BC}$ and$\overline{AB}$ is a transversal, so by postulate 10 which states that two angles represent the corresponding angles only when a separate line travels through these two lines which are having a parallel relation between them.

Write the corresponding angles.

$\angle x\cong \angle DAB$

That is,$\angle x=\angle DAB=65°$

Step 2. Determine the value of angle y

Also since ABCD is a parallelogram, so by theorem 5-2 which states that the relationship between two angles that are related to a parallelogram and positioned on the opposite sides of each other is congruent.

Write the opposite angles of provided figure.

$\angle y\cong \angle DAB$

That is,$\angle y=\angle DAB=65°$

Step 3. Determine the value of angle z

Now, lines$\overline{AB}\parallel \overline{CD}$ and$\overline{AD}$ is a transversal, so by theorem 3-3 which states that relationship between two angles is supplementary when these angles are positioned on the same side of the interior of two lines which are having a separate line that is travelling through them,

Write the supplementary angles.

$\angle ADC+\angle DAB=180°$

$\begin{array}{c}z+65°=180°\\ z=180°-65°\\ z=115°\end{array}$