Question

two parallel lines, u and v, are cut by the transversal f as shown.
suppose ( m angle 1 = 50 ^ { circ } ).
complete the statements below.
we see that ( angle 1 ) and ( angle 2 ) are (choose one)
thus, ( angle 1 ) and ( angle 2 ) are (choose one)
so, ( m angle 2 = square ^ { circ } ).
we see that ( angle 2 ) and ( angle 3 ) are (choose one)
and since the lines u and v are parallel, ( angle 2 ) and ( angle 3 ) are (choose one)
so, ( m angle 3 = square ^ { circ } ).
therefore, ( angle 1 ) and ( angle 3 ) are (choose one)
we also see that ( angle 1 ) and ( angle 3 ) are (choose one)
the relationship between ( angle 1 ) and ( angle 3 ) is an example of the following rule.
when parallel lines are cut by a transversal, (choose one)

Explanation:

Step1: Identify the relationship between ∠1 and ∠2

∠1 and ∠2 are adjacent angles that form a linear pair.

Step2: Calculate the measure of ∠2

Since ∠1 and ∠2 are supplementary (sum to \(180^{\circ}\)), \(m\angle2 = 180^{\circ}-m\angle1\). Substituting \(m\angle1 = 50^{\circ}\), we get \(m\angle2=180 - 50=130^{\circ}\).

Step3: Identify the relationship between ∠2 and ∠3

∠2 and ∠3 are alternate interior angles (as lines \(u\) and \(v\) are parallel and \(f\) is the transversal).

Step4: Calculate the measure of ∠3

Since ∠2 and ∠3 are congruent (alternate interior angles theorem), \(m\angle3=m\angle2 = 130^{\circ}\).

Step5: Identify the relationship between ∠1 and ∠3

∠1 and ∠3 are supplementary (as \(m\angle1 + m\angle3=50 + 130 = 180^{\circ}\)) and they are also same - side exterior and interior angles (when considering the parallel lines \(u\) and \(v\) and transversal \(f\)).

Answer:

1. ∠1 and ∠2 are "a linear pair".
2. ∠1 and ∠2 are "supplementary".
3. \(m\angle2 = 130^{\circ}\).
4. ∠2 and ∠3 are "alternate interior angles".
5. ∠2 and ∠3 are "congruent".
6. \(m\angle3 = 130^{\circ}\).
7. ∠1 and ∠3 are "supplementary".
8. ∠1 and ∠3 are "same - side exterior and interior angles".
9. When parallel lines are cut by a transversal, "same - side exterior and interior angles are supplementary".