Chapter 3.3, Problem 10ST1

To determine

To identify: all pairs of angles that must be congruent given that $\overline{ED}\parallel \overline{AC}$ .

Answer to Problem 10ST1

  $\angle 8\cong \angle 2$

  $\angle 4\cong \angle 7$

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.

Consider the given figure:

It can be observed that $\angle 8\text{ and }\angle 2$ are two non adjacent interior angles which are on the opposite sides of the transversal $\overline{EB}$ .

Similarly, $\angle 7\text{ and }\angle 4$ are two non adjacent interior angles which are on the opposite sides of the transversal $\overline{DB}$ .

Thus, by using the above definitions, it can be said that $\angle 8\text{ and }\angle 2$ and $\angle 7\text{ and }\angle 4$ are alternate interior angles.

So by Theorem 3-2, $\angle 8\cong \angle 2$ and $\angle 4\cong \angle 7$ .

Now, the given figure has no corresponding angles so no more congruent angles.