Chapter 3.3, Problem 12ST1

To determine

To find: $m\angle 8$ and $m\angle EDC$ given that $m\angle 2=65$ .

Answer to Problem 12ST1

  $m\angle 8=65$

  $m\angle EDC=115$

Explanation of Solution

Given:

The diagram:

Concept Used:

Corresponding angles: These are the two angles which relative to the two lines are in the corresponding position.

Postulate 10: If two parallel lines are cut by a transversal then the corresponding angles are congruent.

Alternate interior angles: These are the two non adjacent interior angles which are on the opposite sides of the transversal.

Theorem 3-2: If two parallel lines are cut by a transversal then the alternate interior angles are congruent.

Theorem 3-3: If two parallel lines are cut by a transversal then the same-side interior angles are supplementary.

Consider the given figure:

In this figure, lines $\overline{ED}\parallel \overline{AC}$ and $\overline{EB}$ is a transversal, so by theorem 3-2,

  $\angle 2\cong \angle 8$ (Alternate-interior angles)

This implies that,

  $m\angle 8=m\angle 2=65$

Similarly, lines $\overline{EB}\parallel \overline{DC}$ and $\overline{BC}$ is a transversal, so by postulate 10,

  $\angle 5\cong \angle 2$ (Corresponding angles)

This implies that,

  $m\angle 5=m\angle 2=65$

Now, lines $\overline{EB}\parallel \overline{DC}$ and $\overline{DC}$ is a transversal, so by theorem 3-3, $\angle EDC$ and $\angle 5$ are supplementary, that is,

  $\begin{array}{c}m\angle EDC+m\angle 5=180\\ m\angle EDC+65=180\\ m\angle EDC=180-65\\ m\angle EDC=115\end{array}$