Question

1. which angles would the alternate interior angles theorem state are congruent?

∠2 and ∠8, ∠3 and ∠5
∠2 and ∠3, ∠5 and ∠8
∠1 and ∠3, ∠5 and ∠7
∠2 and ∠4, ∠6 and ∠8

Explanation:

Step1: Recall alternate interior angles definition

Alternate interior angles are non - adjacent angles that lie between two lines and on opposite sides of a transversal.

Step2: Analyze each option

• For option 1:
• $\angle2$ and $\angle8$: $\angle2$ is above the transversal \(t\) and \(\angle8\) is below the transversal \(t\), and they are between the two parallel lines.
• $\angle3$ and $\angle5$: $\angle3$ is below the transversal \(t\) and \(\angle5\) is above the transversal \(t\), and they are between the two parallel lines.
• For option 2: $\angle2$ and $\angle3$ are adjacent angles (linear pair), and $\angle5$ and $\angle8$ are adjacent angles (linear pair), not alternate interior angles.
• For option 3: $\angle1$ and $\angle3$ are not between the two parallel lines ( $\angle1$ is outside the region between the two parallel lines), and $\angle5$ and $\angle7$ are adjacent angles (linear pair).
• For option 4: $\angle2$ and $\angle4$ are vertical angles, and $\angle6$ and $\angle8$ are vertical angles, not alternate interior angles.

Answer:

$\angle2$ and $\angle8$, $\angle3$ and $\angle5$ (the first option)