Question

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the figure shows a transversal that intersects two parallel lines. none of the angles are right angles. which angles are congruent to ∠3? select all that apply. ∠1 ∠2 ∠4 ∠5 ∠6 ∠7 ∠8 remember, corresponding angles are congruent and vertical angles are congruent.

Explanation:

Step1: Identify vertical angle to ∠3

Vertical angles are congruent. ∠3 and ∠2 are not vertical angles; ∠3 and ∠6? No, vertical angle of ∠3 is ∠2? No, correction: ∠3's vertical angle is ∠2? No, looking at the diagram: ∠3 and ∠2 are adjacent, vertical angle of ∠3 is ∠6? No, wait: vertical angles are opposite each other when two lines intersect. ∠3 is opposite ∠2? No, no: the top parallel line intersected by transversal: ∠1, ∠2, ∠3, ∠4. ∠3 is opposite ∠2? No, ∠1 is opposite ∠4, ∠3 is opposite ∠2? No, correction: ∠3 and ∠2 are supplementary, vertical angle of ∠3 is ∠6? No, corresponding angles: ∠3 corresponds to ∠7? No, wait: corresponding angles are in same relative position on parallel lines. ∠3 is top-left on the top parallel line, so corresponding angle is top-left on bottom parallel line: ∠7? No, no: ∠1 is top-right, corresponding to ∠5; ∠2 is bottom-right, corresponding to ∠6; ∠3 is top-left, corresponding to ∠7; ∠4 is bottom-left, corresponding to ∠8. Vertical angle of ∠3: when two lines intersect, ∠3 is opposite ∠2? No, no: the transversal intersects top line: ∠1 (top right), ∠2 (bottom right), ∠3 (top left), ∠4 (bottom left). So ∠1 and ∠4 are vertical, ∠3 and ∠2 are vertical? No, that can't be, ∠3 and ∠2 are adjacent. Oh right! I messed up: ∠1 and ∠4 are vertical (opposite), ∠3 and ∠2 are vertical? No, no, ∠3 is adjacent to ∠1 and ∠4. So vertical angle of ∠3 is ∠2? No, no, vertical angles are equal, so ∠3 ≅ ∠2? No, that's not right. Wait the hint says corresponding angles and vertical angles are congruent. So first, vertical angle of ∠3: ∠3 and ∠2? No, no, let's reorient:

Top parallel line, transversal cuts it:
- Upper right: ∠1
- Lower right: ∠2
- Upper left: ∠3
- Lower left: ∠4

Bottom parallel line, transversal cuts it:
- Upper right: ∠5
- Lower right: ∠6
- Upper left: ∠7
- Lower left: ∠8

So vertical angles: ∠1 ≅ ∠4, ∠3 ≅ ∠2? No, ∠3 and ∠2 are adjacent, they form a linear pair, so they are supplementary. Oh! I made a mistake: ∠1 and ∠4 are vertical, ∠3 and ∠2 are not. Wait no, ∠3 is opposite ∠2? No, ∠3 is opposite ∠6? No, ∠3 is on top line left upper, ∠6 is bottom line right lower. Corresponding angles: ∠3 (top left upper) corresponds to ∠7 (bottom left upper) → so ∠3 ≅ ∠7. Vertical angle of ∠3: wait, no, ∠3's vertical angle is ∠2? No, that's adjacent. Wait no, the two lines are the top parallel line and the transversal. So when transversal intersects top line, the four angles are ∠1, ∠2, ∠3, ∠4. So ∠1 is opposite ∠4, ∠3 is opposite ∠2. Oh right! I confused the positions. So ∠3 and ∠2 are vertical angles? No, that would mean they are equal, but they are adjacent. No, no, vertical angles are non-adjacent, opposite each other. So ∠1 is opposite ∠4, ∠3 is opposite ∠2? No, that can't be, ∠3 is next to ∠1. I think I labeled wrong: let's correct:

When a transversal cuts a line, the four angles are:
- Top right: ∠1
- Top left: ∠2
- Bottom left: ∠3
- Bottom right: ∠4

No, the diagram shows ∠1 top right, ∠3 top left, ∠4 bottom left, ∠2 bottom right on the top line. So ∠1 (top right) and ∠4 (bottom left) are vertical? No, that's diagonal. Oh! I see my mistake: vertical angles are opposite each other, so ∠1 (top right) is opposite ∠2 (bottom right)? No, no, vertical angles are across the vertex. So the intersection of transversal and top line: the two pairs of vertical angles are (∠1, ∠4) and (∠3, ∠2). Yes, that's correct: ∠1 and ∠4 are opposite, ∠3 and ∠2 are opposite, so they are vertical angles, hence congruent.

Then corresponding angles: ∠3 (top left on top line) corresponds to ∠7 (top left on bottom line), so ∠3 ≅ ∠7. Also, ∠3's vertical angle is ∠2? No, wait ∠3 ≅ ∠2 (vertical), and ∠2's corresponding angle is ∠6, so ∠2 ≅ ∠6, so ∠3 ≅ ∠6? No, no, corresponding angle of ∠3 is ∠7, corresponding angle of ∠2 is ∠6. Wait the hint says corresponding angles are congruent, vertical angles are congruent.

So let's list:
1. Vertical angle of ∠3: ∠2? No, wait no, ∠3 and ∠2 are adjacent, they form a linear pair, so they add to 180. I was wrong! Vertical angles are non-adjacent. So ∠1 and ∠4 are vertical (non-adjacent, opposite), ∠3 and ∠2 are non-adjacent? No, ∠3 is next to ∠1 and ∠4. Oh my god, I messed up the labeling. Let's look at the diagram again:

Top line: transversal cuts it, the angle top right is ∠1, bottom right is ∠2, top left is ∠3, bottom left is ∠4. So the two pairs of vertical angles are ∠1 & ∠4, ∠3 & ∠2. Yes, ∠3 and ∠2 are opposite each other across the intersection, so they are vertical angles, hence congruent. Then corresponding angles: ∠3 is top left on top line, so corresponding angle is top left on bottom line, which is ∠7, so ∠3 ≅ ∠7. Also, ∠2 (vertical to ∠3) has corresponding angle ∠6, so ∠2 ≅ ∠6, so ∠3 ≅ ∠6? No, that would be alternate interior? Wait no, alternate interior angles are also congruent, but the hint only mentions vertical and corresponding. Wait the question says "Remember, corresponding angles are congruent and vertical angles are congruent." So we only use those two rules.

So:
- Vertical angle of ∠3: ∠2? No, wait ∠3 and ∠2 are vertical, so ∠3 ≅ ∠2.
- Corresponding angle of ∠3: ∠7, so ∠3 ≅ ∠7.
- Also, ∠2's corresponding angle is ∠6, so ∠2 ≅ ∠6, so ∠3 ≅ ∠6 (transitive), but is that allowed? Wait no, the hint says vertical and corresponding are congruent, so we can use vertical to get ∠3 ≅ ∠2, then corresponding to get ∠2 ≅ ∠6, so ∠3 ≅ ∠6. Also, ∠3's corresponding angle is ∠7, so ∠3 ≅ ∠7. Wait no, wait let's do it properly:

Step1: Identify vertical angle to ∠3

Vertical angles are congruent. ∠3 and ∠2 are not vertical, wait no: ∠3 is opposite ∠2? No, ∠3 is top left, ∠2 is bottom right, so they are vertical angles (opposite each other at the intersection). So $\angle 3 \cong \angle 2$.

Step2: Identify corresponding angle to ∠3

Corresponding angles on parallel lines are congruent. ∠3 is top-left on the top parallel line, so its corresponding angle is top-left on the bottom parallel line, which is $\angle 7$. So $\angle 3 \cong \angle 7$.

Step3: Use transitivity with vertical/corresponding

∠2 (congruent to ∠3) has a corresponding angle ∠6, so $\angle 2 \cong \angle 6$, so $\angle 3 \cong \angle 6$.

Wait no, wait another way: ∠3's alternate exterior angle is ∠6? No, the hint only says vertical and corresponding. Wait no, let's check again:

Wait the diagram:
Top line, transversal:
- ∠1: top right
- ∠2: bottom right
- ∠3: top left
- ∠4: bottom left

Bottom line, transversal:
- ∠5: top right
- ∠6: bottom right
- ∠7: top left
- ∠8: bottom left

So corresponding angles:
∠1 ↔ ∠5, ∠2 ↔ ∠6, ∠3 ↔ ∠7, ∠4 ↔ ∠8. So these are congruent.

Vertical angles:
∠1 ↔ ∠4, ∠3 ↔ ∠2, ∠5 ↔ ∠8, ∠7 ↔ ∠6. These are congruent.

So ∠3 is congruent to ∠2 (vertical) and ∠7 (corresponding), and since ∠2 ≅ ∠6 (corresponding), then ∠3 ≅ ∠6. Wait but ∠3 ≅ ∠2 (vertical), ∠2 ≅ ∠6 (corresponding), so ∠3 ≅ ∠6. Also, ∠3 ≅ ∠7 (corresponding), and ∠7 ≅ ∠6 (vertical)? No, ∠7 and ∠6 are adjacent, supplementary. Wait no, ∠7's vertical angle is ∠6? No, ∠7 is top left on bottom line, vertical angle is ∠6? No, ∠7's vertical angle is ∠6? No, ∠7 and ∠6 are adjacent, so they are linear pair. Oh right! ∠7's vertical angle is ∠6? No, ∠5 and ∠8 are vertical, ∠7 and ∠6 are vertical? No, that's wrong. ∠7 is top left, ∠6 is bottom right, so they are vertical angles, yes! Oh right! I messed up again. On the bottom line, ∠7 (top left) and ∠6 (bottom right) are vertical angles, so ∠7 ≅ ∠6. So ∠3 ≅ ∠7 (corresponding), ∠7 ≅ ∠6 (vertical), so ∠3 ≅ ∠6.

Wait but let's list all congruent angles to ∠3:
1. ∠2 (vertical to ∠3)
2. ∠7 (corresponding to ∠3)
3. ∠6 (since ∠7 ≅ ∠6 vertical, so ∠3 ≅ ∠6)

Wait but let's check the options: the options are ∠1, ∠2, ∠4, ∠5, ∠6, ∠7, ∠8.

Wait but let's go back to the hint: "Remember, corresponding angles are congruent and vertical angles are congruent." So we can use these two rules to find all:

- ∠3 ≅ ∠2 (vertical angles)
- ∠3 ≅ ∠7 (corresponding angles)
- ∠2 ≅ ∠6 (corresponding angles), so ∠3 ≅ ∠6 (transitive, using vertical and corresponding)
- ∠7 ≅ ∠6 (vertical angles), so ∠3 ≅ ∠6 (transitive)

Wait but is ∠3 congruent to ∠6? Yes, they are alternate exterior angles, which are congruent, and that can be proven via vertical and corresponding angles.

Wait but let's check linear pairs: ∠3 + ∠1 = 180, so ∠3 is not congruent to ∠1. ∠3 + ∠4 = 180, so not congruent to ∠4. ∠5 is corresponding to ∠1, which is supplementary to ∠3, so ∠5 is supplementary to ∠3, not congruent. ∠8 is corresponding to ∠4, which is supplementary to ∠3, so not congruent.

So the angles congruent to ∠3 are ∠2, ∠6, ∠7.

Wait but let's verify again:
- Vertical angle of ∠3: ∠2 → congruent.
- Corresponding angle of ∠3: ∠7 → congruent.
- Corresponding angle of ∠2: ∠6 → congruent, so ∠3 ≅ ∠6.
- Vertical angle of ∠7: ∠6 → congruent, so ∠3 ≅ ∠6.

Yes, that's correct.

Answer:

∠2, ∠6, ∠7