Question

1. in the figure below, the angles are formed by a transversal and two parallel lines.

which statement best explains why ∠7 and ∠1 are congruent?

they are supplementary angles.

they are alternate interior angles.

they are corresponding angles.

they are alternate exterior angles.

clear all

Explanation:

To determine why ∠7 and ∠1 are congruent, we analyze the angle relationships formed by a transversal and two parallel lines:

1. Supplementary Angles: These are angles that add up to 180°, which does not explain congruence (equal measure) unless they are both 90°, so this is incorrect.
2. Alternate Interior Angles: These lie between the two parallel lines and on opposite sides of the transversal. ∠7 and ∠1 do not fit this description.
3. Corresponding Angles: These are in the same position relative to the parallel lines and the transversal. ∠7 and ∠1 are not in corresponding positions.
4. Alternate Exterior Angles: These lie outside the two parallel lines and on opposite sides of the transversal. Wait, no—actually, ∠7 and ∠1: let's re-examine. Wait, maybe I made a mistake. Wait, let's look at the diagram. Wait, the two parallel lines are the horizontal ones? Wait, no, the diagram has two parallel lines cut by a transversal. Wait, ∠1 and ∠7: let's check their positions. Wait, maybe the correct relationship is alternate interior angles? Wait, no, the options: wait, the correct answer is "They are alternate interior angles"? Wait, no, let's recall:

Wait, alternate interior angles are congruent when lines are parallel. Let's identify the lines: the two parallel lines (let's say the lower and upper horizontal lines) and the transversal (the slanted line). ∠1 is between the two parallel lines, on one side of the transversal. ∠7 is also between the two parallel lines, on the opposite side of the transversal. So they are alternate interior angles. Wait, but the options include "They are alternate interior angles" as a choice. Wait, the question is "Which statement best explains why ∠7 and ∠1 are congruent?" So the correct answer is "They are alternate interior angles" because alternate interior angles formed by a transversal cutting parallel lines are congruent.

Wait, let's re-express:

• Supplementary angles: sum to 180°, not congruent (unless both 90°), so incorrect.
• Alternate interior angles: lie between the parallel lines, on opposite sides of the transversal, and are congruent. ∠1 and ∠7 fit this: between the two parallel lines, on opposite sides of the transversal. So this is correct.
• Corresponding angles: same position relative to parallel lines and transversal. ∠1 and ∠7 are not in corresponding positions.
• Alternate exterior angles: lie outside the parallel lines, which ∠1 and ∠7 do not (they are between the lines). So incorrect.

Thus, the correct statement is "They are alternate interior angles."

Answer:

They are alternate interior angles.