Question

given that the two horizontal lines are parallel, identify all the numbered angles that are congruent to the given angle. list the numbers of any angles that are congruent to the given angle. (use a comma to separate answers as needed. do not include the $angle$ symbol in your answer.)

Explanation:

Step1: Identify given angle property

The given angle (∠2) is 96°, so congruent angles also measure 96°.

Step2: Use vertical angles theorem

Vertical angles are congruent. ∠1 is vertical to ∠2? No, ∠1 is supplementary to ∠2. ∠4 is vertical to ∠1? No, ∠5 is vertical to ∠2's corresponding angle. Wait, correct: ∠1 is supplementary to ∠2 (180-96=84°). ∠2's vertical angle is ∠1? No, no: ∠2 and ∠1 are adjacent on a line, so ∠1=84°. ∠2's corresponding angle is ∠6? No, parallel lines: corresponding angles are congruent. ∠2 corresponds to ∠6, so ∠6=96°. Vertical angle of ∠2 is ∠1? No, no: ∠2 and ∠1 are linear pair, vertical angle of ∠2 is ∠1? No, vertical angles are opposite: ∠2 and ∠1 are not opposite, ∠2's vertical angle is ∠1? Wait no, the intersection: top line, ∠2 and ∠1 are adjacent, ∠2's vertical angle is ∠1? No, ∠3 is vertical to ∠2? No, ∠3 is supplementary to ∠2. Wait, let's reorient:
- On the top parallel line, the transversal creates ∠2 (96°), its vertical angle is ∠1? No, ∠1 is 180-96=84°, so vertical angle of ∠2 is ∠1? No, that's wrong. Vertical angles are opposite, so ∠2 and ∠1 are not opposite: ∠2's vertical angle is ∠1? No, ∠2 and ∠3 are a linear pair? No, ∠2 and ∠1 are a linear pair, ∠3 and ∠2 are vertical? No, no, the top intersection: angles 1,2,3, and the unlabeled one. ∠2 and ∠1 are adjacent, so ∠1=84°, ∠3 is vertical to ∠2, so ∠3=96°? No, no, vertical angles are equal, so if ∠2=96°, its vertical angle is ∠3? No, that can't be, because ∠2 and ∠3 are on the same side? Wait, no, the transversal crosses the top horizontal line: the four angles are ∠1, ∠2, ∠3, and an unlabeled one. ∠2 and ∠1 are adjacent (linear pair), so ∠1=180-96=84°. ∠2's vertical angle is the unlabeled one, so ∠3 is vertical to ∠1, so ∠3=84°. Then, for the bottom parallel line: corresponding angles to ∠2 is ∠6, so ∠6=96°. ∠6's vertical angle is ∠5, so ∠5=96°. ∠4 is corresponding to ∠1, so ∠4=84°, ∠7 is vertical to ∠6? No, ∠7 is vertical to ∠4? No, ∠7 is adjacent to ∠6, so ∠7=84°.
Wait, correct congruent angles to ∠2 (96°) are ∠3? No, no, I messed up. Let's do it properly:
1. Vertical angles: ∠2 and ∠3? No, ∠2 and the angle opposite to it (unlabeled) are vertical, so ∠3 is adjacent to ∠2, so ∠3=84°. Wait, no, the given angle is ∠2=96°, so its vertical angle is the angle opposite, which is unlabeled. Then, corresponding angles: ∠2 corresponds to ∠6, so ∠6=96°. ∠6's vertical angle is ∠5, so ∠5=96°. Also, alternate interior angles: ∠2 and ∠7? No, ∠2 and ∠7 are same-side interior, so supplementary. ∠2 and ∠5 are alternate interior, so ∠5=96°. ∠6 is corresponding to ∠2, so ∠6=96°. Also, vertical angle of ∠6 is ∠5, so ∠5=96°. Wait, and what about ∠3? No, ∠3 is 84°, same as ∠1, ∠4, ∠7.
Wait, no, let's start over:

Step1: Define congruent angle rules

Congruent angles have equal measure. For parallel lines cut by transversal:
- Vertical angles are congruent
- Corresponding angles are congruent
- Alternate interior/exterior angles are congruent

Step2: Identify given angle measure

Given ∠2 = 96°

Step3: Find corresponding angles

∠2 corresponds to ∠6, so ∠6 = 96°

Step4: Find vertical angles

Vertical angle of ∠6 is ∠5, so ∠5 = 96°

Step5: Find alternate interior angles

∠2's alternate interior angle is ∠5, which we already have. Also, is there another? ∠2's alternate exterior angle is ∠... no, ∠3 is 84°, ∠1 is 84°, ∠4 is 84°, ∠7 is 84°.
Wait, no, wait: ∠2 and ∠3 are not vertical, ∠2 and the angle opposite to it (unlabeled) are vertical, so ∠3 is adjacent to ∠2, so ∠3=84°. So the congruent angles to ∠2 (96°) are ∠5, ∠6. Wait, no, wait another way: ∠2 and ∠1 are linear pair, so ∠1=84°, ∠3 is vertical to ∠1, so ∠3=84°. ∠4 is corresponding to ∠1, so ∠4=84°, ∠7 is vertical to ∠4, so ∠7=84°. ∠6 is corresponding to ∠2, so ∠6=96°, ∠5 is vertical to ∠6, so ∠5=96°. Also, ∠2's alternate interior angle is ∠5, which is 96°, and ∠6 is corresponding, so yes, ∠5 and ∠6 are congruent to ∠2.

Wait, but wait, is ∠3 congruent? No, ∠3 is 84°, which is supplementary. So the angles congruent to ∠2 (96°) are ∠5, ∠6.

Wait, no, wait, maybe I mixed up the labels. Let's look at the diagram: top horizontal line, transversal crosses it, angles labeled 1,2,3. ∠2 is 96°, so ∠1 is adjacent to ∠2, so ∠1=84°, ∠3 is opposite to ∠2, so ∠3=96° (vertical angle). Oh! I made a mistake here. Vertical angles are opposite, so ∠2 and ∠3 are vertical? No, no, if the transversal crosses the line, the four angles are: top-right ∠1, top-left ∠2, bottom-left ∠3, bottom-right unlabeled. So ∠2 and ∠3 are vertical? No, ∠2 and the bottom-right are vertical, ∠1 and ∠3 are vertical. Oh! That's my mistake. So ∠2 (top-left) and bottom-right (unlabeled) are vertical, so they are 96°. ∠1 (top-right) and ∠3 (bottom-left) are vertical, so 84°. Then, for the bottom parallel line: angles 4,5,6,7. ∠4 is top-left, ∠5 is bottom-left, ∠6 is top-right, ∠7 is bottom-right. Corresponding angles: ∠2 (top-left of top line) corresponds to ∠4 (top-left of bottom line)? No, no, corresponding angles are same position: ∠2 (top-left of top intersection) corresponds to ∠6 (top-left of bottom intersection)? No, no, the transversal is the same, so ∠2 (top-left, top line) corresponds to ∠6 (top-left, bottom line)? No, ∠6 is top-right of bottom line. Wait, no, corresponding angles are in the same relative position: ∠2 is on the top line, left side of transversal, above the line? No, ∠2 is 96°, so it's obtuse. ∠6 is on the bottom line, right side of transversal, above the line, which is corresponding to ∠2, so ∠6=96°. ∠5 is on the bottom line, left side of transversal, below the line, which is alternate interior to ∠2, so ∠5=96°. ∠3 is on the top line, left side of transversal, below the line, which is same-side interior to ∠2, so supplementary, 84°. ∠1 is on the top line, right side of transversal, above the line, which is supplementary to ∠2, 84°. ∠4 is on the bottom line, right side of transversal, below the line, corresponding to ∠1, 84°. ∠7 is on the bottom line, left side of transversal, above the line, corresponding to ∠3, 84°.

Ah, now I see. So ∠2=96°, its corresponding angle is ∠6=96°, its alternate interior angle is ∠5=96°, and its vertical angle is the unlabeled angle (not numbered). So the numbered angles congruent to ∠2 are ∠5, ∠6.

Wait, but wait, is there another? ∠3 is 84°, no. ∠1 is 84°, no. ∠4 is 84°, no. ∠7 is 84°, no. So yes, ∠5 and ∠6.

Wait, no, wait another way: vertical angles of ∠6 is ∠5? No, ∠6 and ∠5 are adjacent, linear pair? No, ∠6 and ∠7 are linear pair, ∠6 and ∠5 are vertical? No, ∠6 and ∠5 are on the same side? No, bottom line: transversal crosses it, angles 4 (top-left), 6 (top-right), 7 (bottom-right), 5 (bottom-left). So ∠6 (top-right) and ∠5 (bottom-left) are vertical angles, so they are equal, 96°. ∠4 (top-left) and ∠7 (bottom-right) are vertical angles, 84°.

Yes, so ∠2=96°, corresponding angle is ∠6=96°, vertical angle of ∠6 is ∠5=96°, so ∠5 and ∠6 are congruent to ∠2.

Answer:

5, 6