Question

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

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 angles are congruent.

Step2: Analyze $\angle 3$'s Vertical Angle

$\angle 3$ and $\angle 1$ are vertical angles, so $\angle 3 \cong \angle 1$.

Step3: Analyze Corresponding Angles

The two parallel lines are cut by transversals. For the first transversal (the left one), $\angle 3$ and $\angle 7$ are corresponding angles (since the lines are parallel), so $\angle 3 \cong \angle 7$. For the second transversal (the right one), $\angle 7$ and $\angle 5$ are vertical angles, and since $\angle 3 \cong \angle 7$, then $\angle 3 \cong \angle 5$ (by transitivity, as $\angle 7 \cong \angle 5$).

Step4: Check Other Angles

$\angle 2$ is supplementary to $\angle 3$ (linear pair), so not congruent. $\angle 4$ is supplementary to $\angle 3$ (linear pair), not congruent. $\angle 6$: Let's see, $\angle 5$ and $\angle 6$ are supplementary, so $\angle 6$ is supplementary to $\angle 5$, hence supplementary to $\angle 3$, not congruent. $\angle 8$: $\angle 7$ and $\angle 8$ are supplementary, so $\angle 8$ is supplementary to $\angle 7$, hence supplementary to $\angle 3$, not congruent. So the congruent angles to $\angle 3$ are $\angle 1$, $\angle 5$, $\angle 7$.

Answer:

$\angle 1$, $\angle 5$, $\angle 7$