Question

the figure shows a transversal that intersects two parallel lines. none of the angles are right angle
complete the sentence.
∠4 and ∠8 are corresponding angles.
what are all the angles in the figure that are congruent to ∠2?
∠1, ∠7, ∠8
∠3, ∠5, ∠8
∠3, ∠6, ∠7
∠4, ∠6, ∠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 - interior and alternate - exterior angles are congruent.
- Vertical angles: $\angle2$ and $\angle4$ are vertical angles? Wait, no. Wait, $\angle2$ and $\angle4$: Wait, let's look at the intersection points. At the top intersection, the angles are $\angle1$, $\angle2$, $\angle3$, $\angle4$. At the bottom intersection, the angles are $\angle5$, $\angle6$, $\angle7$, $\angle8$.
- $\angle2$ and $\angle6$: Corresponding angles (since the two lines are parallel and the transversal cuts them, $\angle2$ and $\angle6$ are in the same relative position). So $\angle2\cong\angle6$.
- $\angle2$ and $\angle8$: Wait, no. Wait, $\angle2$ and $\angle4$: Vertical angles? Wait, $\angle1$ and $\angle3$ are vertical angles, $\angle2$ and $\angle4$ are vertical angles? Wait, no, at the top intersection, the vertical angles: $\angle1$ and $\angle3$ are vertical? Wait, no, when two lines intersect, vertical angles are opposite each other. So at the top intersection, the two lines intersect, so $\angle1$ and $\angle3$ are vertical? No, $\angle1$ and $\angle4$? Wait, maybe I made a mistake. Let's re - establish:
- At the top intersection (of the two lines, not the transversal and parallel lines), the angles: if we have two lines intersecting, $\angle1$ and $\angle3$ are vertical angles, $\angle2$ and $\angle4$ are vertical angles. But then the transversal cuts the two parallel lines. So the parallel lines are the ones with the angles $\angle1 - \angle4$ and $\angle5 - \angle8$? Wait, no, the two parallel lines are the ones that are cut by the transversal. So the transversal is the line that goes from the top intersection to the bottom intersection, and the two parallel lines are the other two lines (one with $\angle1 - \angle4$ and one with $\angle5 - \angle8$).
- So, $\angle2$ and $\angle6$: corresponding angles (same position relative to the parallel lines and transversal), so $\angle2\cong\angle6$.
- $\angle2$ and $\angle8$: Wait, no. Wait, $\angle4$ and $\angle8$ are corresponding angles (as given in the first part: $\angle4$ and $\angle8$ are corresponding angles). Since $\angle2$ and $\angle4$ are vertical angles (at the top intersection, the two lines intersect, so $\angle2$ and $\angle4$ are vertical angles, so $\angle2\cong\angle4$). And since $\angle4\cong\angle8$ (corresponding angles), then $\angle2\cong\angle8$. Also, $\angle2$ and $\angle4$ are vertical angles, so $\angle2\cong\angle4$, and $\angle4$ and $\angle8$ (corresponding), $\angle2$ and $\angle6$ (corresponding). Wait, let's check the options.
- The options are:
- Option 1: $\angle1,\angle7,\angle8$
- Option 2: $\angle3,\angle5,\angle8$
- Option 3: $\angle3,\angle6,\angle7$
- Option 4: $\angle4,\angle6,\angle8$
- Let's analyze each angle:
- $\angle2$ and $\angle4$: vertical angles, so $\angle2\cong\angle4$.
- $\angle2$ and $\angle6$: corresponding angles, so $\angle2\cong\angle6$.
- $\angle2$ and $\angle8$: Since $\angle4\cong\angle8$ (corresponding angles) and $\angle2\cong\angle4$ (vertical angles), by transitivity, $\angle2\cong\angle8$.

Step2: Check the Options

- Option 1: $\angle1,\angle7,\angle8$: $\angle1$ is supplementary to $\angle2$ (since they form a linear pair), so $\angle1
ot\cong\angle2$. So this option is wrong.
- Option 2: $\angle3,\angle5,\angle8$: $\angle3$ is supplementary to $\angle2$ (linear pair), so $\angle3
ot\cong\angle2$. So this option is wrong.
- Option 3: $\angle3,\angle6,\angle7$: $\angle3$ is supplementary to $\angle2$, $\angle7$: let's see, $\angle7$ and $\angle5$ are vertical angles, $\angle5$ and $\angle3$: maybe alternate - interior? No, $\angle7$ is not congruent to $\angle2$. So this option is wrong.
- Option 4: $\angle4,\angle6,\angle8$: $\angle4\cong\angle2$ (vertical angles), $\angle6\cong\angle2$ (corresponding angles), $\angle8\cong\angle2$ (since $\angle4\cong\angle8$ (corresponding) and $\angle2\cong\angle4$ (vertical), so $\angle2\cong\angle8$). So this option is correct.

Answer:

$\angle4$, $\angle6$, $\angle8$ (the option with $\angle4$, $\angle6$, $\angle8$)