Question

in the diagram, m ∥ n and m and n are cut by transversal p. which angle pairs are congruent? check all that apply. □ ∠1 and ∠5 □ ∠3 and ∠8 □ ∠6 and ∠4 □ ∠8 and ∠2 □ ∠4 and ∠7

Explanation:

Step1: Recall Angle Theorems

When two parallel lines are cut by a transversal, corresponding angles, alternate interior angles, and alternate exterior angles are congruent, and vertical angles are congruent. Also, corresponding angles (same position relative to parallel lines and transversal) are congruent, alternate interior (between lines, opposite sides of transversal) are congruent, alternate exterior (outside lines, opposite sides) are congruent, and vertical angles (opposite each other when two lines intersect) are congruent.

Step2: Analyze ∠1 and ∠5

∠1 and ∠5 are corresponding angles (same position relative to parallel lines \(m \parallel n\) and transversal \(p\)). Corresponding angles are congruent when lines are parallel. So ∠1 ≅ ∠5.

Step3: Analyze ∠3 and ∠8

∠3 and ∠8: Let's check their positions. ∠3 is between \(m\) and \(n\) (interior), ∠8 is also interior but on same side? Wait, no. Wait, ∠3 and ∠8: ∠3 is on line \(n\), ∠8 is on line \(m\). Wait, actually, ∠3 and ∠8: are they alternate interior? No, ∠3 and ∠8: let's see, transversal \(p\) cuts \(m\) and \(n\). ∠3 is adjacent to ∠2, ∠8 is adjacent to ∠7. Wait, maybe I made a mistake. Wait, ∠3 and ∠8: are they same - side interior? No, same - side interior angles are supplementary, not congruent (unless lines are perpendicular, which they aren't here). So ∠3 and ∠8 are not congruent.

Step4: Analyze ∠6 and ∠4

∠6 and ∠4: ∠6 is on line \(m\), ∠4 is on line \(n\). Let's see, ∠6 and ∠2 are vertical angles (∠6 ≅ ∠2), and ∠2 and ∠4 are vertical angles? No, ∠2 and ∠4: wait, ∠2 and ∠4 are vertical angles? Wait, the intersection of the two non - parallel (the other two) lines: when two lines intersect, vertical angles are congruent. Wait, the two lines (not \(m\) and \(n\)) intersect, forming ∠1, ∠2, ∠3, ∠4. So ∠2 ≅ ∠4 (vertical angles). And ∠6 ≅ ∠2 (corresponding angles, since \(m \parallel n\) and transversal \(p\)). So by transitivity, ∠6 ≅ ∠4. Wait, or maybe ∠6 and ∠4: ∠6 is corresponding to ∠2, ∠2 is vertical to ∠4, so ∠6 ≅ ∠4. So ∠6 and ∠4 are congruent.

Step5: Analyze ∠8 and ∠2

∠8 and ∠2: ∠8 and ∠2 are alternate interior angles? Wait, \(m \parallel n\), transversal \(p\). ∠2 is on line \(n\), ∠8 is on line \(m\), between the lines (interior), and on opposite sides of transversal \(p\). So alternate interior angles. Alternate interior angles are congruent when lines are parallel. So ∠8 ≅ ∠2.

Step6: Analyze ∠4 and ∠7

∠4 and ∠7: ∠4 is on line \(n\), ∠7 is on line \(m\). Let's see, ∠4 and ∠1 are vertical angles (∠4 ≅ ∠1), ∠1 and ∠5 are corresponding (∠1 ≅ ∠5), ∠5 and ∠7 are vertical angles (∠5 ≅ ∠7). So by transitivity, ∠4 ≅ ∠7? Wait, no. Wait, ∠4 and ∠7: ∠4 is on the lower - left, ∠7 is on the lower - right. Wait, maybe ∠4 and ∠7 are corresponding? No. Wait, ∠4 and ∠7: let's check the positions. ∠4 is outside the two parallel lines (exterior), ∠7 is inside? No, ∠7 is between \(m\) and \(p\)? Wait, no. Wait, \(m\) and \(n\) are parallel, transversal \(p\). ∠4 is on line \(n\), below the transversal, ∠7 is on line \(m\), below the transversal. Wait, no, ∠7 is on the right - hand side of transversal \(p\), ∠4 is on the left - hand side. Wait, maybe ∠4 and ∠7 are not congruent. Wait, let's re - check. ∠4 and ∠7: ∠4 is vertical to ∠1, ∠1 and ∠5 are corresponding, ∠5 and ∠7 are vertical. So ∠4 ≅ ∠1, ∠1 ≅ ∠5, ∠5 ≅ ∠7, so ∠4 ≅ ∠7. Wait, that's correct. Wait, but earlier when we checked ∠3 and ∠8, we thought they were not congruent. Wait, maybe I made a mistake there. Wait, no, let's go back.

Wait, let's re - list the angle pairs:

1. ∠1 and ∠5: corresponding angles, congruent. Correct.

2. ∠3 and ∠8: Let's see, ∠3 is on line \(n\), between \(n\) and transversal \(p\), ∠8 is on line \(m\), between \(m\) and transversal \(p\), same side? No, ∠3 and ∠8: ∠3 is adjacent to ∠2, ∠8 is adjacent to ∠7. Wait, maybe ∠3 and ∠8 are same - side interior? Same - side interior angles are supplementary, so they are not congruent (unless each is 90 degrees, which they aren't). So ∠3 and ∠8 are not congruent.

3. ∠6 and ∠4: As we saw, ∠6 ≅ ∠2 (vertical), ∠2 ≅ ∠4 (vertical), so ∠6 ≅ ∠4. Correct.

4. ∠8 and ∠2: Alternate interior angles, so congruent. Correct.

5. ∠4 and ∠7: ∠4 ≅ ∠1 (vertical), ∠1 ≅ ∠5 (corresponding), ∠5 ≅ ∠7 (vertical), so ∠4 ≅ ∠7. Correct? Wait, no, ∠5 and ∠7 are vertical angles? Wait, ∠5 and ∠7: when transversal \(p\) intersects line \(m\), the angles formed are ∠5, ∠6, ∠7, ∠8. So ∠5 and ∠7 are vertical angles? No, ∠5 and ∠7 are adjacent, forming a linear pair? Wait, no, ∠5 and ∠6 are adjacent (linear pair), ∠6 and ∠7 are adjacent (linear pair), ∠7 and ∠8 are adjacent (linear pair), ∠8 and ∠5 are adjacent (linear pair)? No, that's not right. Wait, when two lines intersect, they form two pairs of vertical angles. So when transversal \(p\) intersects line \(m\), the intersection creates ∠5, ∠6, ∠7, ∠8. So ∠5 and ∠7 are vertical angles? No, ∠5 and ∠7 are opposite each other? Wait, no, ∠5 is top - left, ∠7 is bottom - right. Wait, no, when two lines intersect, vertical angles are opposite each other. So if we have two intersecting lines, say line \(m\) and transversal \(p\), the vertical angles are ∠5 and ∠7, ∠6 and ∠8. Ah! There we go. So ∠5 ≅ ∠7 (vertical angles), ∠6 ≅ ∠8 (vertical angles). So ∠5 ≅ ∠7. Then ∠1 ≅ ∠5 (corresponding), ∠4 ≅ ∠1 (vertical), so ∠4 ≅ ∠5, and ∠5 ≅ ∠7, so ∠4 ≅ ∠7. So that's correct.

Wait, but earlier when we thought ∠3 and ∠8: ∠3 is on line \(n\), ∠8 is on line \(m\). ∠3 and ∠8: are they alternate interior? No, ∠3 is between \(n\) and \(p\), ∠8 is between \(m\) and \(p\), same side? No, ∠3 is on the left - hand side of transversal \(p\), ∠8 is on the right - hand side? Wait, no, the transversal \(p\) is a line going from top - left to bottom - right? Wait, the diagram: line \(p\) is a transversal, with angles 5,6 at the top, 8,7 at the bottom. Line \(m\) and \(n\) are parallel, going from top - left to bottom - right (with arrows). The other two lines (the non - parallel ones) intersect at a point, forming angles 1,2,3,4. So ∠3 is on line \(n\), below the transversal \(p\), ∠8 is on line \(m\), below the transversal \(p\). Wait, ∠3 and ∠8: are they corresponding? ∠3 is on line \(n\), ∠8 is on line \(m\), same position relative to transversal \(p\) (below the transversal, on the right - hand side? Wait, maybe I messed up the diagram orientation. Anyway, according to the angle theorems:

- Corresponding angles: congruent (∠1 and ∠5: correct)
- Alternate interior angles: congruent (∠8 and ∠2: correct, since ∠2 is on line \(n\), ∠8 on line \(m\), between the lines, opposite sides of transversal)
- Vertical angles: congruent (∠6 and ∠4: ∠6 ≅ ∠2 (vertical), ∠2 ≅ ∠4 (vertical), so ∠6 ≅ ∠4: correct)
- ∠4 and ∠7: ∠4 ≅ ∠1 (vertical), ∠1 ≅ ∠5 (corresponding), ∠5 ≅ ∠7 (vertical), so ∠4 ≅ ∠7: correct)
- ∠3 and ∠8: not congruent (same - side interior or not corresponding/alternate)

Answer:

The congruent angle pairs are:

• ∠1 and ∠5
• ∠6 and ∠4
• ∠8 and ∠2
• ∠4 and ∠7

So the checked boxes should be for ∠1 and ∠5, ∠6 and ∠4, ∠8 and ∠2, ∠4 and ∠7.