Question

decide whether each pair of angles is congruent or not congruent.
angles | congruent or not congruent
∠2 and ∠5 | not congruent
∠2 and ∠7 | ?
options: congruent, not congruent

Explanation:

Step1: Identify Angle Relationships

In the diagram, we have two transversals cutting a horizontal line. ∠2 and ∠7: Let's check their positions. ∠2 and ∠5 are not congruent (given), but for ∠2 and ∠7, we can see that the lines forming ∠2 and ∠7—wait, actually, looking at the vertical angles or corresponding angles? Wait, no, let's see the angles. Wait, ∠2 and ∠7: are they alternate interior or something? Wait, no, actually, ∠2 and ∠7—wait, maybe they are congruent? Wait, no, wait. Wait, the two transversals: the first transversal (left) and the second transversal (right). ∠2 and ∠7: let's check their measures. Wait, ∠2 and ∠4 are vertical angles, ∠5 and ∠7 are vertical angles? Wait, no, ∠5 and ∠7: ∠5 and ∠7 are adjacent? Wait, no, the horizontal line is cut by two transversals. So ∠2 and ∠7: are they congruent? Wait, actually, ∠2 and ∠7—wait, maybe I made a mistake. Wait, the first transversal (left) creates angles 1,2,3,4. The second transversal (right) creates 5,6,7,8. Now, ∠2 and ∠7: let's see, ∠2 and ∠5: not congruent (given). Now, ∠2 and ∠7: are they congruent? Wait, no, wait. Wait, maybe ∠2 and ∠7 are congruent? Wait, no, wait. Wait, maybe the lines are not parallel? Wait, the diagram: the two transversals—are they parallel? Wait, the left transversal and the right transversal: do they look parallel? The arrows on the transversals: the left transversal has an arrow going up and down, the right transversal also has arrows. Wait, maybe the two transversals are parallel? Wait, if the two transversals are parallel, then ∠2 and ∠7 would be congruent? Wait, no, maybe not. Wait, no, let's think again. Wait, ∠2 and ∠7: let's check their measures. Wait, ∠2 and ∠4 are vertical angles (so equal). ∠5 and ∠7 are vertical angles (so equal). But ∠2 and ∠5 are not congruent (given), so ∠2 ≠ ∠5, so ∠2 ≠ ∠7 (since ∠5 = ∠7). Wait, that makes sense. So ∠2 and ∠7: not congruent? Wait, no, wait, maybe I messed up. Wait, the given says ∠2 and ∠5 are not congruent. So ∠2 ≠ ∠5. And ∠5 and ∠7 are vertical angles? Wait, no, ∠5 and ∠7: ∠5 is adjacent to ∠7? Wait, no, ∠5 and ∠7: ∠5 and ∠7 are supplementary? Wait, no, ∠5 and ∠7: if the horizontal line is straight, then ∠5 + ∠7 = 180? No, ∠5 and ∠7: ∠5 and ∠7 are adjacent angles forming a linear pair? Wait, no, ∠5 and ∠7: the horizontal line is cut by the right transversal, so ∠5 and ∠7 are vertical angles? Wait, no, vertical angles are opposite each other. So ∠5 and ∠7: no, ∠5 and ∠7 are adjacent, with ∠5 + ∠7 = 180? Wait, no, ∠5 and ∠6 are adjacent, ∠6 and ∠8, ∠8 and ∠7, ∠7 and ∠5. Wait, no, the right transversal: angles 5 (top left), 6 (top right), 7 (bottom left), 8 (bottom right). So ∠5 and ∠7 are vertical angles? No, ∠5 and ∠7: ∠5 is top left, ∠7 is bottom left—so they are vertical angles? Wait, no, vertical angles are opposite, so ∠5 and ∠7: no, ∠5 and ∠7 are adjacent, forming a linear pair? Wait, no, a linear pair is adjacent and supplementary. So ∠5 + ∠7 = 180? No, ∠5 and ∠6 are a linear pair (supplementary), ∠6 and ∠8, ∠8 and ∠7, ∠7 and ∠5. Wait, no, ∠5 and ∠7: if the transversal is straight, then ∠5 and ∠7 are vertical angles? Wait, no, vertical angles are formed by two intersecting lines. So the horizontal line and the right transversal intersect, forming ∠5, ∠6, ∠7, ∠8. So ∠5 and ∠7 are vertical angles? No, ∠5 and ∠7: ∠5 is (top, left), ∠7 is (bottom, left)—so they are vertical angles? Wait, no, vertical angles are opposite, so ∠5 (top left) and ∠7 (bottom left) are not opposite. Wait, ∠5 (top left) and ∠8 (bottom right) are vertical angles? No, ∠5 and ∠8: no, ∠5 and ∠7: adjacent. Wait, I think I made a mistake. Let's correct: when two lines intersect (horizontal and right transversal), the vertical angles are ∠5 and ∠8, ∠6 and ∠7. Oh! Yes, that's right. So ∠5 and ∠8 are vertical angles (equal), ∠6 and ∠7 are vertical angles (equal). Similarly, for the left transversal: ∠1 and ∠4 are vertical angles, ∠2 and ∠3 are vertical angles. Now, ∠2 and ∠7: ∠2 is from the left transversal, ∠7 is from the right transversal (∠7 = ∠6, since ∠6 and ∠7 are vertical angles? Wait, no, ∠6 and ∠7 are vertical angles? Wait, no, ∠6 and ∠7: if the right transversal intersects the horizontal line, then ∠6 (top right) and ∠7 (bottom left) are adjacent, forming a linear pair? No, I'm confused. Let's start over.

Two lines: horizontal (let's call it line H) and two transversals: left transversal (L) and right transversal (R).

Intersection of L and H: angles 1 (top left), 2 (top right), 3 (bottom left), 4 (bottom right). So ∠1 = ∠4 (vertical), ∠2 = ∠3 (vertical).

Intersection of R and H: angles 5 (top left), 6 (top right), 7 (bottom left), 8 (bottom right). So ∠5 = ∠8 (vertical), ∠6 = ∠7 (vertical).

Now, ∠2 (top right of L-H intersection) and ∠7 (bottom left of R-H intersection). Now, are ∠2 and ∠7 congruent?

Given that ∠2 and ∠5 (top left of R-H intersection) are not congruent (given: ∠2 and ∠5 are not congruent). Now, ∠7 = ∠6 (vertical angles, ∠6 and ∠7 are vertical? Wait, no, ∠6 (top right of R-H) and ∠7 (bottom left of R-H) are adjacent, forming a linear pair (supplementary), so ∠6 + ∠7 = 180. Wait, no, vertical angles are opposite, so when R and H intersect, the vertical angles are ∠5 (top left) and ∠8 (bottom right), ∠6 (top right) and ∠7 (bottom left). Wait, no, that's not right. Vertical angles are formed by two intersecting lines, so the opposite angles. So if R and H intersect, the four angles are: top left (∠5), top right (∠6), bottom left (∠7), bottom right (∠8). So the vertical angles are ∠5 (top left) and ∠8 (bottom right), ∠6 (top right) and ∠7 (bottom left). Yes, that's correct. So ∠5 = ∠8, ∠6 = ∠7.

Now, ∠2 (top right of L-H) and ∠7 (bottom left of R-H, which is equal to ∠6, since ∠6 = ∠7). Now, ∠2 and ∠5: not congruent (given). ∠5 and ∠6: are they congruent? If L and R are parallel, then ∠2 and ∠6 would be corresponding angles (congruent), but since ∠2 and ∠5 are not congruent, and ∠5 and ∠6 are supplementary (since they are adjacent on a straight line: ∠5 + ∠6 = 180), so ∠2 ≠ ∠5, so ∠2 ≠ 180 - ∠6. But ∠7 = ∠6, so ∠2 ≠ 180 - ∠7. Wait, this is getting too complicated. Wait, the problem says "Decide whether each pair of angles is congruent or not congruent." For ∠2 and ∠7: looking at the diagram, the two transversals (L and R) are not parallel (since ∠2 and ∠5 are not congruent, which would be corresponding angles if L and R were parallel). So since L and R are not parallel, ∠2 and ∠7 are not congruent? Wait, but the options are congruent or not congruent. Wait, maybe I made a mistake. Wait, let's look at the diagram again. The left transversal and the right transversal: the left transversal has an arrow going up and down, the right transversal also has an arrow. Wait, maybe the transversals are parallel? Wait, the arrows on the transversals: if the transversals are parallel, then ∠2 and ∠7 would be congruent? Wait, no, if the transversals are parallel, then ∠2 and ∠7 would be alternate interior angles? Wait, no, the horizontal line is the transversal for the two parallel transversals. Wait, I'm getting confused. Let's go back to the given: ∠2 and ∠5 are not congruent. Now, ∠2 and ∠7: are they congruent? Wait, maybe the answer is not congruent? Wait, no, wait. Wait, maybe ∠2 and ∠7 are congruent. Wait, I think I messed up. Wait, let's check the vertical angles. ∠2 and ∠3 are vertical, ∠7 and ∠6 are vertical. Now, ∠2 and ∠7: if the two transversals are parallel, then ∠2 and ∠6 would be corresponding angles (congruent), so ∠2 = ∠6, and since ∠6 = ∠7 (vertical angles), then ∠2 = ∠7. But the problem says ∠2 and ∠5 are not congruent. If the transversals are parallel, ∠2 and ∠5 would be corresponding angles (congruent), but they are not, so the transversals are not parallel. Therefore, ∠2 and ∠7 are not congruent. Wait, but that contradicts. Wait, maybe the diagram shows the transversals as parallel? Let me look at the diagram again. The left transversal and the right transversal: the arrows on the transversals—are they parallel? The left transversal has an arrow going up and down, the right transversal also has an arrow. Maybe the transversals are parallel. Then, if transversals are parallel, and the horizontal line is a transversal, then ∠2 and ∠7 would be alternate interior angles? Wait, no, alternate interior angles are between the two parallel lines. Wait, the horizontal line is cut by two parallel transversals. So ∠2 and ∠7: are they alternate interior angles? No, alternate interior angles would be between the parallel transversals. Wait, maybe I'm overcomplicating. The problem gives ∠2 and ∠5 as not congruent. Now, ∠2 and ∠7: let's see, ∠7 is equal to ∠6 (vertical angles), and ∠2 and ∠5: not congruent. If the transversals are parallel, ∠2 and ∠5 would be congruent (corresponding angles), but they are not, so transversals are not parallel. Therefore, ∠2 and ∠7 are not congruent. So the answer is not congruent.

Step2: Confirm the Relationship

Since ∠2 and ∠5 are not congruent (given) and ∠7 is related to ∠5 (via the right transversal's angle relationships), and the transversals are not parallel (as ∠2 and ∠5, which would be corresponding angles if parallel, are not congruent), ∠2 and ∠7 do not have equal measures. Thus, they are not congruent.

Answer:

not congruent