Question

the figure shows a pair of parallel lines cut by a transversal. which three angles are congruent to ∠2? choose all three correct angles. ∠4 ∠6 ∠8 what argument can you make to explain why m∠2 = m∠8? complete the statement. when a transversal cuts parallel lines, the? angles are congruent. adjacent alternate interior same - side interior

Explanation:

Sub - question 1: Which three angles are congruent to ∠2?

Step 1: Recall angle relationships with parallel lines and transversal

When two parallel lines are cut by a transversal, vertical angles are congruent, corresponding angles are congruent, and alternate - exterior/interior angles are congruent.

• Vertical angles: ∠2 and ∠4 are vertical angles, so ∠2 ≅ ∠4.
• Corresponding angles: ∠2 and ∠6 are corresponding angles (since the lines are parallel), so ∠2 ≅ ∠6.
• Alternate - exterior angles: ∠2 and ∠8, let's see the positions. ∠2 is an upper - right angle at the first intersection, ∠8 is a lower - left angle at the second intersection. Since the lines are parallel, alternate - exterior angles are congruent, so ∠2 ≅ ∠8.

Step 2: Identify the three angles

From the above analysis, the three angles congruent to ∠2 are ∠4, ∠6, and ∠8.

Step 1: Recall angle types

• Adjacent angles: These are angles that share a common side and vertex, and they are supplementary (sum to 180°) in the case of linear pairs, so they are not congruent in general (except when they are right angles).
• Alternate interior angles: These are angles between the two parallel lines, on opposite sides of the transversal. ∠2 and ∠8 are not between the two parallel lines, so they are not alternate interior angles.
• Alternate exterior angles: ∠2 and ∠8 are alternate exterior angles. When a transversal cuts parallel lines, alternate exterior angles are congruent.

Step 2: Choose the correct term

Among the options, "alternate interior" is incorrect, "adjacent" is incorrect. The correct term is "alternate exterior" (wait, but in the options given, there is "alternate interior". Wait, maybe I made a mistake. Wait, let's re - examine the positions. The two parallel lines are the two non - horizontal lines. The transversal is the horizontal line. ∠2 is above the transversal and to the right of the first non - horizontal line. ∠8 is below the transversal and to the left of the second non - horizontal line. Wait, maybe they are alternate interior? No, alternate interior angles are between the two parallel lines. Wait, the two parallel lines are the two slanted lines. The transversal is the horizontal line. So the region between the two slanted lines is the interior. ∠2 is above the transversal (exterior to the interior region) and ∠8 is below the transversal (exterior to the interior region) but on the opposite sides of the transversal. Wait, maybe the question has a typo or my initial analysis is wrong. Wait, the options are "adjacent", "alternate interior", "same - side interior". Wait, let's think again. ∠2 and ∠8: when we consider the two parallel lines (slanted) and the transversal (horizontal), ∠2 and ∠8 are alternate interior angles? Wait, no. Wait, the two parallel lines are cut by the transversal. The interior of the two parallel lines is between them. ∠2 is above the transversal (outside the interior) and ∠8 is below the transversal (outside the interior). Wait, maybe the question meant alternate interior. Wait, no. Wait, let's look at the positions again. The first intersection: angles 1,2,3,4. The second intersection: angles 5,6,7,8. The two parallel lines are the two lines with arrows (slanted). The transversal is the horizontal line. So ∠2 and ∠8: ∠2 is at the top - right of the first intersection, ∠8 is at the bottom - left of the second intersection. These are alternate exterior angles. But the option given is "alternate interior". Wait, maybe the diagram is different. Wait, maybe the two parallel lines are the horizontal line? No, the horizontal line is the transversal. The two slanted lines are parallel. So the interior of the two slanted lines: between them. ∠3 and ∠5 are alternate interior angles, ∠4 and ∠8? Wait, no. Wait, maybe I messed up. Wait, the correct angle relationship for ∠2 and ∠8: when two parallel lines are cut by a transversal, alternate interior angles are congruent. Wait, maybe ∠2 and ∠8 are alternate interior? No, ∠2 is above the transversal, ∠8 is below the transversal. Wait, perhaps the question has a mistake, but among the options, "alternate interior" is the only one that is a congruent angle relationship when lines are parallel (except adjacent and same - side interior). Wait, same - side interior angles are supplementary. Adjacent angles are supplementary (linear pair) or share a side. So the correct term from the options is "alternate interior". Wait, maybe…

Answer:

∠4, ∠6, ∠8

Sub - question 2: What argument can you make to explain why m∠2 = m∠8? Complete the statement. When a transversal cuts parallel lines, the? angles are congruent.