Question

if possible, write congruence statements for the pairs of corresponding angles in each figure. if it is not possible, leave the answer blank.
move an answer into each space provided.
it is, not possible.
∠1 ≅
∠2 ≅
∠3 ≅
∠4 ≅
is ∠5 ∠6 ∠7 ∠8

Explanation:

Step1: Analyze ∠1

∠1 and ∠3: Wait, no, let's check vertical angles and corresponding angles. Wait, the lines g and h: are they parallel? Wait, the problem says "if possible". But looking at the angles, ∠1 and ∠3? No, wait, ∠1 and ∠5? No, wait, vertical angles: ∠1 and ∠5? No, ∠1 and ∠3? Wait, no, let's see the transversal l. Wait, the two lines g and h, with transversal l. Wait, ∠1 and ∠3: no, maybe the problem is that g and h are not marked as parallel, so we can't assume corresponding angles are congruent. Wait, but the options have "is not" possible? Wait, the first part: "It [is, not] possible." Then the angle congruences. Wait, if we can't assume parallel lines, then we can't write congruence statements for corresponding angles (since corresponding angles are congruent only when lines are parallel, which isn't given here). So first, the "It is not possible" part. Then the angle congruences: but wait, maybe I misread. Wait, the figure: lines g and h, transversal l. Angles 1,2,3,4 above l; 5,6,7,8 below. ∠1 and ∠5: vertical angles? No, ∠1 and ∠5 are adjacent? Wait, no, ∠1 and ∠5: ∠1 and ∠5 are vertical? No, ∠1 and ∠5: ∠1 and ∠5 are supplementary? Wait, no, vertical angles are opposite each other. ∠1 and ∠5: no, ∠1 and ∠5 are adjacent? Wait, line g intersects l at a point, creating ∠1, ∠2, ∠5, ∠6. So ∠1 and ∠5 are vertical? Wait, no, ∠1 and ∠5: ∠1 is above l, left of g; ∠5 is below l, left of g. So they are vertical angles? Wait, vertical angles are congruent. Wait, but the problem is about corresponding angles between the two lines g and h. Corresponding angles would be ∠1 and ∠3, ∠2 and ∠4, ∠5 and ∠7, ∠6 and ∠8. But for corresponding angles to be congruent, lines g and h must be parallel, which isn't given. So we can't assume that. Therefore, it is not possible to write congruence statements for corresponding angles (since we don't know if g || h). So first, "It is not possible." Then the angle congruences: but the blanks for ∠1 ≅, ∠2 ≅, etc. But if it's not possible, maybe the answer is that we can't write them, but the options have "is not" and the angle options. Wait, maybe I made a mistake. Wait, the first part: "It [is, not] possible." Let's check the options. The first dropdown: "is" or "not". Then the angle blanks. If we can't assume parallel lines, then corresponding angles (∠1 and ∠3, ∠2 and ∠4, etc.) aren't necessarily congruent. So "It is not possible." Then the angle congruences: but the options have ∠5, ∠6, ∠7, ∠8. Wait, maybe the problem is that even without parallel lines, vertical angles are congruent. Wait, ∠1 and ∠5: vertical angles? Wait, ∠1 and ∠5: ∠1 is above l, left of g; ∠5 is below l, left of g. So they are vertical angles (opposite each other when two lines intersect), so ∠1 ≅ ∠5. Similarly, ∠2 ≅ ∠6 (vertical angles), ∠3 ≅ ∠7 (vertical angles), ∠4 ≅ ∠8 (vertical angles). But the problem says "pairs of corresponding angles". Corresponding angles are angles in the same relative position at each intersection. So ∠1 (above l, left of g) and ∠3 (above l, left of h) are corresponding angles. But for corresponding angles to be congruent, g || h. Since that's not given, we can't say ∠1 ≅ ∠3. But vertical angles are always congruent, regardless of parallel lines. Wait, maybe the problem is mixing corresponding angles (between the two lines) and vertical angles. But the question says "congruence statements for the pairs of corresponding angles". So corresponding angles between g and h, with transversal l. So ∠1 and ∠3, ∠2 and ∠4, ∠5 and ∠7, ∠6 and ∠8. But without g || h, we can't say they are congruent. Therefore, it is not possible to write congruence statements for corresponding angles (since we don't know if g || h). So the first part: "It is not possible." Then the angle congruences: but the options have ∠5, ∠6, ∠7, ∠8. Wait, maybe the problem has a typo, or I misinterpret. Wait, let's re-express:

1. First, determine if it's possible: since we don't know if g || h, we can't assume corresponding angles are congruent, so "It is not possible."

2. Then, the angle congruences: but the blanks are for ∠1 ≅, ∠2 ≅, ∠3 ≅, ∠4 ≅. But if it's not possible, maybe the answer is that we can't write them, but the options have "is not" and the angles. Wait, maybe the intended answer is that it is not possible, so the angle congruences are blank, but the options have "is not" and the angles. Wait, maybe I made a mistake. Let's check the vertical angles: ∠1 ≅ ∠5 (vertical angles), ∠2 ≅ ∠6 (vertical angles), ∠3 ≅ ∠7 (vertical angles), ∠4 ≅ ∠8 (vertical angles). But the question is about corresponding angles, not vertical angles. Corresponding angles are in the same position relative to the two lines and the transversal. So ∠1 (top-left of g and l) and ∠3 (top-left of h and l) are corresponding angles. But without g || h, they aren't necessarily congruent. So the answer is:

It is not possible. Then ∠1 ≅ (blank, but the options have ∠5, ∠6, ∠7, ∠8? Wait, no, maybe the problem is that the user made a mistake, but according to the options, the first part is "is not", then the angle congruences: but if we consider vertical angles, but the question is about corresponding angles. Wait, maybe the problem is designed to say it's not possible, so the angle congruences are not possible. So:

Step1: Determine Possibility

Since the lines \( g \) and \( h \) are not marked as parallel, we cannot assume corresponding angles are congruent (corresponding angles are congruent only when lines are parallel, a condition not given here). Thus, it is not possible to write congruence statements for corresponding angles.

Step2: Angle Congruences

For corresponding angles (e.g., \( \angle 1 \) and \( \angle 3 \), \( \angle 2 \) and \( \angle 4 \), etc.), we lack the parallel line condition, so no congruence statements can be made for corresponding angles. However, if we mistakenly considered vertical angles (but the question specifies "corresponding angles"), the answer remains that it is not possible.

Answer:

It is not possible.
\( \angle 1 \cong \) (blank, but since it's not possible, no congruence for corresponding angles)
\( \angle 2 \cong \) (blank)
\( \angle 3 \cong \) (blank)
\( \angle 4 \cong \) (blank)

But according to the options, the first part is "is not", and the angle blanks: maybe the intended answer is that it's not possible, so the congruence statements for corresponding angles (∠1 and ∠3, ∠2 and ∠4, etc.) can't be written, but the vertical angles are congruent, but the question is about corresponding angles. Wait, maybe the problem has an error, but based on the given options, the correct selection is:

It is not possible.
\( \angle 1 \cong \) (no corresponding angle congruence, but if we follow the options, maybe the answer is that it's not possible, so the blanks are left, but the first part is "is not".

Final Answer:
It is not possible.
\( \angle 1 \cong \) (blank), \( \angle 2 \cong \) (blank), \( \angle 3 \cong \) (blank), \( \angle 4 \cong \) (blank)

But according to the dropdown options, the first part is "is not", and the angle options are ∠5, ∠6, ∠7, ∠8. Wait, maybe the problem meant vertical angles, but the question says corresponding angles. Alternatively, maybe the lines are parallel (implied by the figure), but not marked. If we assume g || h, then:

∠1 ≅ ∠3 (corresponding angles), ∠2 ≅ ∠4 (corresponding angles), ∠3 ≅ ∠7? No, ∠3 and ∠7 are alternate interior? Wait, no, corresponding angles: ∠1 and ∠3, ∠2 and ∠4, ∠5 and ∠7, ∠6 and ∠8.

If we assume g || h (maybe the figure implies it), then:

∠1 ≅ ∠3, ∠2 ≅ ∠4, ∠3 ≅ ∠7? No, ∠3 and ∠7: no, ∠3 is above l, left of h; ∠7 is below l, left of h. So alternate interior. But the question is about corresponding angles. So corresponding angles: ∠1 (above, left of g) and ∠3 (above, left of h) – corresponding, so ∠1 ≅ ∠3. ∠2 (above, right of g) and ∠4 (above, right of h) – corresponding, so ∠2 ≅ ∠4. ∠3 (above, left of h) and ∠7 (below, left of h)? No, ∠3 and ∠7 are vertical? No, ∠3 and ∠7: ∠3 is above l, ∠7 is below l, same side of h. So supplementary? No, if parallel, ∠3 and ∠7 are same-side interior? No, alternate interior. Wait, I'm confused. Let's start over.

Corresponding angles: when two lines are cut by a transversal, corresponding angles are in the same position relative to each line and the transversal. So for lines g and h, transversal l:

• ∠1 (top-left of g and l) corresponds to ∠3 (top-left of h and l)
• ∠2 (top-right of g and l) corresponds to ∠4 (top-right of h and l)
• ∠5 (bottom-left of g and l) corresponds to ∠7 (bottom-left of h and l)
• ∠6 (bottom-right of g and l) corresponds to ∠8 (bottom-right of h and l)

For these to be congruent, g must be parallel to h (corresponding angles postulate). Since the problem doesn't state g || h, we can't assume that. Therefore, it is not possible to write congruence statements for corresponding angles. Thus:

"It is not possible."

Then, the angle congruences: since we can't assume parallelism, we can't write ∠1 ≅ ∠3, ∠2 ≅ ∠4, etc. So the blanks remain empty, but the options have "is not" and the angle options. Maybe the problem intended vertical angles, but the question says corresponding angles. Given the options, the correct answer is:

It is not possible.
\( \angle 1 \cong \) (no corresponding angle congruence, so blank)
\( \angle 2 \cong \) (blank)
\( \angle 3 \cong \) (blank)
\( \angle 4 \cong \) (blank)

But according to the dropdown, the first part is "is not", and the angle options are ∠5, ∠6, ∠7, ∠8. Maybe the problem has a mistake, but based on the question, the answer is that it's not possible, so the congruence statements for corresponding angles can't be written.

Final Answer:
It is not possible.
\( \angle 1 \cong \) (no answer), \( \angle 2 \cong \) (no answer), \( \angle 3 \cong \) (no answer), \( \angle 4 \cong \) (no answer)

But in the given options, the first part is "is not", and the angle blanks: maybe the intended answer is that it's not possible, so the congruence statements are not possible, hence the blanks are left, and the first part is "is not".