Question

if parallel lines a and b are cut by transversal t, then which statement about angles 1 and 2 must be true?
options:
∠1 and ∠2 are right angles
∠1 is the supplement of ∠2
∠1 ≅ ∠2
∠1 is the complement of ∠2

Explanation:

Step1: Recall Parallel Line Angles

When parallel lines are cut by a transversal, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary. Also, vertical angles are equal. Let's analyze ∠1 and ∠2. ∠1 and the angle adjacent to ∠2 (vertical to ∠1's corresponding angle) – actually, ∠1 and ∠2: let's see the positions. ∠1 and the angle that is vertical to ∠2's adjacent? Wait, no. Let's look at the diagram: line a || line b, transversal t. ∠1 and ∠2: ∠1 and the angle above ∠2 (let's call it ∠3) – ∠1 and ∠3 are corresponding angles (so ∠1 ≅ ∠3), and ∠3 and ∠2 are vertical angles (so ∠3 ≅ ∠2). Therefore, by transitivity, ∠1 ≅ ∠2? Wait, no, wait. Wait, maybe I made a mistake. Wait, let's re-examine. Wait, ∠1 and ∠2: let's see the angles. Wait, ∠1 and the angle that is supplementary to ∠2? No, wait. Wait, the correct approach: when two parallel lines are cut by a transversal, corresponding angles are congruent, alternate interior are congruent, consecutive interior are supplementary. Now, ∠1 and ∠2: let's see their positions. ∠1 is on line a, above the transversal, ∠2 is on line b, below the transversal? Wait, no, the diagram: line a is top, line b is bottom. Transversal t crosses them. ∠1 is at the intersection of t and a, above a? Wait, no, the arrows: line a has left arrow and right arrow, so it's horizontal. Transversal t is a line with an upward arrow. ∠1 is between t and a, on the left side? Wait, maybe ∠1 and ∠2 are corresponding angles? Wait, no, maybe ∠1 and ∠2 are congruent because ∠1 and the angle vertical to ∠2 are corresponding, so ∠1 ≅ ∠2? Wait, no, let's check the options. The options are: right angles (no, unless transversal is perpendicular, which isn't stated), supplement (sum to 180), congruent, complement (sum to 90). Wait, the correct answer: ∠1 ≅ ∠2? Wait, no, wait. Wait, maybe I messed up. Wait, let's think again. When parallel lines are cut by a transversal, corresponding angles are congruent. ∠1 and ∠2: are they corresponding? Let's see: ∠1 is at the top intersection, ∠2 is at the bottom intersection. If we look at the transversal, ∠1 and ∠2: ∠1 and the angle that is vertical to ∠2 (let's say ∠2's vertical angle is ∠3) – ∠1 and ∠3 are corresponding angles (since a || b, transversal t), so ∠1 ≅ ∠3. And ∠3 ≅ ∠2 (vertical angles), so ∠1 ≅ ∠2. Wait, but another way: ∠1 and ∠2 – are they alternate exterior or interior? Wait, maybe the correct option is ∠1 ≅ ∠2? Wait, no, wait the options: the third option is ∠1 ≅ ∠2. Wait, but let's check the other options. "∠1 is the supplement of ∠2" – would that be if they are consecutive interior? But consecutive interior are on the same side. ∠1 and ∠2: are they on the same side? No, ∠1 is above line a, ∠2 is below line b? Wait, maybe I got the diagram wrong. Wait, the diagram: line a is top, line b is bottom, both horizontal (left and right arrows). Transversal t is a line with an upward arrow, crossing both. ∠1 is at the intersection of t and a, on the left side (between t and a, above a? No, the angle labeled 1 is at the top intersection, above line a? Wait, no, the arrow on line a: left and right, so line a is horizontal. Transversal t is a line going up, so at the top intersection, ∠1 is on the left side of t, above line a? Wait, no, the angle is between t and a, on the left. Then line b is below, with ∠2 at the intersection of t and b, on the right side? Wait, no, the label 2 is at the bottom intersection, below line b? Wait, maybe ∠1 and ∠2 are corresponding angles. Corresponding angles are congruent when lines are parallel. So ∠1 ≅ ∠2. Wait, but let's check the options. The third option is ∠1 ≅ ∠2. Let's eliminate the others: right angles – no, unless transversal is perpendicular, not given. Supplement – sum to 180, but if they are congruent, sum would be 2∠1, which is 180 only if ∠1 is 90, not given. Complement – sum to 90, same issue. So the correct answer is ∠1 ≅ ∠2? Wait, no, wait I think I made a mistake. Wait, maybe ∠1 and ∠2 are supplementary? Wait, no, let's look at the diagram again. Wait, the transversal t: at the top intersection, ∠1 is adjacent to a vertical angle, and at the bottom intersection, ∠2 is adjacent to that vertical angle's corresponding angle. Wait, maybe the correct answer is ∠1 ≅ ∠2. Wait, the options: third option is ∠1 ≅ ∠2. So that's the answer.

Step2: Eliminate Incorrect Options

- "∠1 and ∠2 are right angles": No info that transversal is perpendicular, so false.
- "∠1 is the supplement of ∠2": Supplements sum to 180°, but no info angles sum to 180° (unless consecutive interior, but ∠1 and ∠2 aren't consecutive interior).
- "∠1 is the complement of ∠2": Complements sum to 90°, no info for that.
- "∠1 ≅ ∠2": By corresponding angles (∠1 and ∠2's vertical angle are corresponding, so ∠1 ≅ ∠2's vertical angle, and vertical angles are congruent, so ∠1 ≅ ∠2).

Answer:

$\angle 1 \cong \angle 2$ (the third option: $\boldsymbol{\angle 1 \cong \angle 2}$)