Question

the figure shows a pair of parallel lines cut by a transversal. which three angles are congruent to ∠2? choose all three correct angles. ∠3 ∠4 ∠5 ∠6 ∠7 ∠8

Explanation:

Step1: Recall Angle Relationships

When two parallel lines are cut by a transversal, vertical angles are congruent, and corresponding angles are congruent. Also, alternate - exterior and alternate - interior angles are congruent.
- Vertical angles: $\angle2$ and $\angle4$ are vertical angles, so $\angle2\cong\angle4$.
- Corresponding angles: Since the lines are parallel, $\angle2$ and $\angle6$ are corresponding angles, so $\angle2\cong\angle6$.
- Alternate - exterior angles: $\angle2$ and $\angle8$: Wait, no, let's re - examine. Wait, $\angle2$ and $\angle8$? No, let's look at the vertical angles and corresponding angles again.
Wait, $\angle2$ and $\angle4$ (vertical angles), $\angle2$ and $\angle6$ (corresponding angles), and $\angle2$ and $\angle8$? No, wait, $\angle2$ and $\angle8$? Wait, no, let's look at the other angle. Wait, $\angle2$ and $\angle8$? Wait, no, $\angle2$ and $\angle4$ (vertical), $\angle2$ and $\angle6$ (corresponding), and $\angle2$ and $\angle8$? Wait, no, maybe I made a mistake. Wait, the two parallel lines: the first intersection has angles 1,2,3,4 and the second has 5,6,7,8.
$\angle2$ and $\angle4$ are vertical angles (so congruent). $\angle2$ and $\angle6$ are corresponding angles (since the lines are parallel, corresponding angles are congruent). $\angle2$ and $\angle8$? Wait, no, $\angle6$ and $\angle8$ are vertical angles. Wait, $\angle2$ and $\angle8$: no, let's check $\angle2$ and $\angle8$: $\angle2$ and $\angle8$ are alternate - exterior? Wait, no, the transversal cuts the two parallel lines. So $\angle2$ and $\angle8$: no, $\angle2$ and $\angle6$ (corresponding), $\angle2$ and $\angle4$ (vertical), and $\angle2$ and $\angle8$? Wait, no, $\angle3$ and $\angle5$ are corresponding? Wait, maybe I messed up. Wait, let's list the angle relationships:
1. Vertical angles: $\angle1\cong\angle3$, $\angle2\cong\angle4$, $\angle5\cong\angle7$, $\angle6\cong\angle8$.
2. Corresponding angles: Since the lines are parallel, $\angle2\cong\angle6$ (corresponding), $\angle3\cong\angle7$, $\angle1\cong\angle5$, $\angle4\cong\angle8$.
3. Alternate - interior angles: $\angle3\cong\angle5$, $\angle4\cong\angle6$.
4. Alternate - exterior angles: $\angle2\cong\angle8$, $\angle1\cong\angle7$.

Wait, so $\angle2$ is congruent to $\angle4$ (vertical), $\angle6$ (corresponding), and $\angle8$ (alternate - exterior)? Wait, but the options are $\angle3$, $\angle4$, $\angle5$, $\angle6$, $\angle7$, $\angle8$. Wait, maybe I mis - identified the angles. Let's re - label:
At the first intersection (left), the angles around the transversal and the first parallel line: angle 1 (top - left), angle 2 (top - right), angle 3 (bottom - right), angle 4 (bottom - left). At the second intersection (right), angle 5 (top - left), angle 6 (top - right), angle 7 (bottom - right), angle 8 (bottom - left).
So $\angle2$ (top - right of first intersection) and $\angle4$ (bottom - left of first intersection) are vertical angles (congruent). $\angle2$ (top - right of first intersection) and $\angle6$ (top - right of second intersection) are corresponding angles (congruent, since lines are parallel). $\angle2$ (top - right of first intersection) and $\angle8$ (bottom - left of second intersection): are they congruent? $\angle6$ and $\angle8$ are vertical angles, so $\angle6\cong\angle8$, and since $\angle2\cong\angle6$, then $\angle2\cong\angle8$. Wait, but the options: let's check the options again. The options are $\angle3$, $\angle4$, $\angle5$, $\angle6$, $\angle7$, $\angle8$.
Wait, maybe I made a mistake. Wait, $\angle2$ and $\angle4$ (vertical), $\angle2$ and $\angle6$ (corresponding), and $\angle2$ and $\angle8$? Wait, no, $\angle2$ and $\angle8$: is $\angle8$ congruent to $\angle2$? Let's see: $\angle2$ and $\angle6$ (corresponding), $\angle6$ and $\angle8$ (vertical), so $\angle2\cong\angle8$. Also, $\angle2$ and $\angle4$ (vertical), $\angle2$ and $\angle6$ (corresponding). Wait, but the question says three angles. Wait, maybe $\angle4$, $\angle6$, and $\angle8$? Wait, but $\angle8$ is an option. Wait, let's check the angles again.
Wait, $\angle2$: adjacent to $\angle1$ and $\angle3$. $\angle2$ and $\angle4$ are vertical (so congruent). $\angle2$ and $\angle6$ are corresponding (since the two lines are parallel, the transversal cuts them, so corresponding angles are congruent). $\angle2$ and $\angle8$: since $\angle6$ and $\angle8$ are vertical, and $\angle2\cong\angle6$, then $\angle2\cong\angle8$. So the three angles congruent to $\angle2$ are $\angle4$, $\angle6$, and $\angle8$? Wait, but let's check the options. The options are $\angle3$, $\angle4$, $\angle5$, $\angle6$, $\angle7$, $\angle8$. Wait, maybe I mis - identified the angles. Wait, maybe the two parallel lines: the first line is the one with angles 1 - 4, the second with 5 - 8. So $\angle2$ and $\angle6$ (corresponding), $\angle2$ and $\angle4$ (vertical), and $\angle2$ and $\angle8$? Wait, but $\angle8$ is an option. Wait, but let's check the angle $\angle8$: $\angle8$ and $\angle6$ are vertical, $\angle6$ and $\angle2$ are corresponding, so $\angle8\cong\angle2$. So the three angles are $\angle4$, $\angle6$, $\angle8$? Wait, but let's check the options. The options are $\angle3$, $\angle4$, $\angle5$, $\angle6$, $\angle7$, $\angle8$. So $\angle4$, $\angle6$, $\angle8$ are the three angles. Wait, but let's confirm:
- Vertical angles: $\angle2\cong\angle4$.
- Corresponding angles: $\angle2\cong\angle6$ (since lines are parallel).
- $\angle6\cong\angle8$ (vertical angles), so $\angle2\cong\angle8$ (transitive property).

So the three angles congruent to $\angle2$ are $\angle4$, $\angle6$, and $\angle8$.

Step2: Identify the Angles

From the above analysis, the angles congruent to $\angle2$ are $\angle4$, $\angle6$, and $\angle8$.

Answer:

$\angle4$, $\angle6$, $\angle8$