Question

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 s and t 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 always congruent. So, if we assume the measure of ∠1 is \(x\), then \(m\angle2 = m\angle1\).

Step2: Identify angle - pair relationship of ∠2 and ∠3

Since lines \(s\) and \(t\) are parallel and \(w\) is a transversal, ∠2 and ∠3 are corresponding angles. Corresponding angles formed by parallel lines cut by a transversal are congruent. So, \(m\angle3=m\angle2\).

Step3: Determine relationship between ∠1 and ∠3

Since \(m\angle1 = m\angle2\) and \(m\angle2=m\angle3\), then \(m\angle1 = m\angle3\). ∠1 and ∠3 are congruent. Also, ∠1 and ∠3 are non - adjacent and on the same side of the transversal, so they are corresponding angles.

Answer:

We see that ∠1 and ∠2 are vertical angles. Thus, ∠1 and ∠2 are congruent. So, \(m\angle2=m\angle1\) (let's assume \(m\angle1 = 50^{\circ}\), then \(m\angle2 = 50^{\circ}\)). We see that ∠2 and ∠3 are corresponding angles. And since the lines \(s\) and \(t\) are parallel, ∠2 and ∠3 are congruent. So, \(m\angle3 = m\angle2=50^{\circ}\). Therefore, ∠1 and ∠3 are congruent. We also see that ∠1 and ∠3 are corresponding angles. The relationship between ∠1 and ∠3 is an example of the following rule. When parallel lines are cut by a transversal, corresponding angles are congruent.