Chapter 2.4, Problem 21CE

a.

To determine

To deduce: from the statement $\angle 2\cong \angle 3$

Answer to Problem 21CE

Lines are parallel that is $m\parallel n$

Explanation of Solution

Given information: $\angle 2\cong \angle 3$

Consider the following figure.

  

If a transversal intersects two lines such that the pair of alternate interior angles are congruent then the lines are said to be parallel.

Here, $\angle 2,\angle 3$ are alternate interior angles such that $\angle 2\cong \angle 3$ .

So,

  $m\parallel n$

b.

To determine

To prove: the lines are parallel that is $m\parallel n$

Explanation of Solution

Given information: $\angle 2\cong \angle 3$

Consider the following figure.

  

  $\angle 1\cong \angle 2$ ( $\angle 1,\angle 2$ are vertically opposite angles and vertically opposite angles are always equal)

Also, $\angle 2\cong \angle 3$ (given)

According to transitive property of congruence, things which are congruent to the same thing are congruent to each other0

As $\angle 1\cong \angle 2,\angle 2\cong \angle 3$ ,

  $\angle 1\cong \angle 3$ (using transitive property of congruence)

Now use the theorem which states that if a transversal intersects two lines such that the pair of corresponding angles are congruent then the lines are said to be parallel.

Here, $\angle 1,\angle 3$ are corresponding angles such that $\angle 1\cong \angle 3$ .

So,

  $m\parallel n$