Question

two parallel lines, m and n, are cut by the transversal d as shown. suppose m∠2 = 135°. complete the statements below. we see that ∠2 and ∠3 are (choose one) and since the lines m and n are parallel, ∠2 and ∠3 are (choose one) so, m∠3 = . we see that ∠1 and ∠2 are (choose one) thus, ∠1 and ∠2 are (choose one) so, m∠1 = . therefore, ∠1 and ∠3 are (choose one) we also see that ∠1 and ∠3 are (choose one) the relationship between ∠1 and ∠3 is an example of the following rule. when parallel lines are cut by a transversal, (choose one)

Explanation:

Step1: Identify angle - pair relationship of ∠2 and ∠3

∠2 and ∠3 are alternate interior angles.

Step2: Use parallel - line property

Since m and n are parallel, alternate interior angles are congruent. So, if \(m\angle2 = 135^{\circ}\), then \(m\angle3=135^{\circ}\).

Step3: Identify angle - pair relationship of ∠1 and ∠2

∠1 and ∠2 are a linear pair.

Step4: Use linear - pair property

A linear pair of angles are supplementary. So \(m\angle1 + m\angle2=180^{\circ}\), and \(m\angle1=180 - 135=45^{\circ}\).

Step5: Analyze ∠1 and ∠3 relationship

∠1 and ∠3 are supplementary because \(m\angle1 + m\angle3=45^{\circ}+135^{\circ}=180^{\circ}\). They are also same - side interior angles. When parallel lines are cut by a transversal, same - side interior angles are supplementary.

Answer:

We see that ∠2 and ∠3 are alternate interior angles.
And since the lines m and n are parallel, ∠2 and ∠3 are congruent. So, \(m\angle3 = 135^{\circ}\).
We see that ∠1 and ∠2 are a linear pair.
Thus, ∠1 and ∠2 are supplementary. So, \(m\angle1 = 45^{\circ}\).
Therefore, ∠1 and ∠3 are supplementary.
We also see that ∠1 and ∠3 are same - side interior angles.
The relationship between ∠1 and ∠3 is an example of the following rule: When parallel lines are cut by a transversal, same - side interior angles are supplementary.