Chapter 3.2, Problem 23WE

a)

To determine

To draw: a four sided figure $ABCD$ with $\overline{AB}\parallel \overline{DC}$ and $\overline{AD}\parallel \overline{BC}$ and to prove that $\angle A\cong \angle C$ .

Explanation of Solution

Consider the following data:

  $\overline{AB}\parallel \overline{DC}$

And

  $\overline{AD}\parallel \overline{BC}$

Consider the following figure $ABCD$ :

  

To prove: $\angle A\cong \angle C$

The proof is as follows:

Statement	Reasons
1. $\overline{AB}\parallel \overline{DC}$ and $\overline{AD}\parallel \overline{BC}$	Given
2. $\angle A$ is supplementary to $\angle C$ ; $\angle C$ is supplementary to $\angle B$	If two parallel lines are cut by a traversal the side by side interior angles are supplementary
3. $\angle A\cong \angle C$	If two angles are supplementary of same angle then both angles are congruent

b)

To determine

Whether $\angle B\cong \angle D$

Explanation of Solution

Consider the following data:

  $\overline{AB}\parallel \overline{DC}$

And

  $\overline{AD}\parallel \overline{BC}$

Consider the following figure $ABCD$ :

  

To prove: $\angle B\cong \angle D$

The proof is as follows:

Statement	Reasons
1. $\overline{AB}\parallel \overline{DC}$ and $\overline{AD}\parallel \overline{BC}$	Given
2. $\angle B$ is supplementary to $\angle C$ ; $\angle C$ is supplementary to $\angle D$	If two parallel lines are cut by a traversal the side by side interior angles are supplementary
3. $\angle B\cong \angle D$	If two angles are supplementary of same angle then both angles are congruent