Question

two parallel lines, p and q, are cut by the transversal d as shown. suppose m∠1 = 65°. complete the statements below. we see that ∠1 and ∠2 are choose one thus, ∠1 and ∠2 are choose one so, m∠2 = °. we see that ∠2 and ∠3 are choose one and since the lines p and q are parallel, ∠2 and ∠3 are choose one so, m∠3 = °. 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 ∠1 and ∠2

∠1 and ∠2 are vertical angles. Vertical angles are congruent.

Step2: Find the measure of ∠2

Since vertical angles are congruent and \(m\angle1 = 65^{\circ}\), then \(m\angle2=m\angle1 = 65^{\circ}\).

Step3: Identify angle - pair relationship of ∠2 and ∠3

∠2 and ∠3 are corresponding angles. When two parallel lines \(p\) and \(q\) are cut by a transversal \(d\), corresponding angles are congruent.

Step4: Find the measure of ∠3

Since ∠2 and ∠3 are corresponding angles and \(m\angle2 = 65^{\circ}\), then \(m\angle3=m\angle2 = 65^{\circ}\).

Step5: Identify angle - pair relationship of ∠1 and ∠3

∠1 and ∠3 are alternate exterior angles. Also, ∠1 and ∠3 are congruent.

Step6: State the rule

When parallel lines are cut by a transversal, alternate exterior angles are congruent.

Answer:

We see that ∠1 and ∠2 are vertical angles.
Thus, ∠1 and ∠2 are congruent.
So, \(m\angle2 = 65^{\circ}\).
We see that ∠2 and ∠3 are corresponding angles.
And since the lines \(p\) and \(q\) are parallel, ∠2 and ∠3 are congruent.
So, \(m\angle3 = 65^{\circ}\).
Therefore, ∠1 and ∠3 are alternate exterior angles.
We also see that ∠1 and ∠3 are congruent.
The relationship between ∠1 and ∠3 is an example of the following rule.
When parallel lines are cut by a transversal, alternate exterior angles are congruent.