Question

select all the pairs of alternate exterior angles.
∠rqt and ∠stq ∠stv and ∠stq
∠stv and ∠rqo ∠utq and ∠stv

Explanation:

Step1: Recall Alternate Exterior Angles

Alternate exterior angles are formed when a transversal crosses two parallel lines (or any two lines). They lie outside the two lines and on opposite sides of the transversal.

Step2: Analyze Each Pair

- **∠RQT and ∠STQ**: These are not exterior angles (they are inside the region between the two lines).
- **∠STV and ∠STQ**: These share a common side and vertex, not alternate exterior.
- **∠STV and ∠RQO**: ∠STV is outside line SU, ∠RQO is outside line PR, and they are on opposite sides of transversal VO. Check positions: transversal VO, lines SU and PR. ∠STV (exterior to SU, left of VO) and ∠RQO (exterior to PR, right of VO) – alternate exterior.
- **∠UTQ and ∠STV**: ∠UTQ is adjacent to ∠STV, not alternate exterior. Wait, re - check: Wait, maybe I made a mistake. Wait, let's re - examine the lines. The two lines are SU (with points S, T, U) and PR (with points P, Q, R). The transversal is VO (with points V, T, Q, O).

Exterior angles: For line SU, exterior angles relative to transversal VO would be ∠STV (above SU, left of VO) and ∠UTQ (below SU, right of VO). For line PR, exterior angles relative to transversal VO would be ∠RQO (below PR, right of VO) and ∠PQV (above PR, left of VO). Wait, maybe the correct pair is ∠STV and ∠RQO? Wait, no, let's use the definition again. Alternate exterior angles: two angles that are outside the two lines, on opposite sides of the transversal.

Wait, another approach: The two lines are SU and PR. Transversal is VO.

- ∠STV: outside SU, left of VO.
- ∠RQO: outside PR, right of VO. So they are alternate exterior.

Wait, but also, let's check the other option. Wait, maybe I misread the options. Wait, the options are:

1. ∠RQT and ∠STQ
2. ∠STV and ∠STQ
3. ∠STV and ∠RQO
4. ∠UTQ and ∠STV

Wait, maybe the correct pair is ∠STV and ∠RQO? Wait, no, let's recall the definition. Alternate exterior angles are formed when a transversal intersects two lines, and the angles are outside the two lines and on alternate sides of the transversal.

So for lines SU and PR, transversal VO:

- ∠STV: outside SU, left of VO.
- ∠RQO: outside PR, right of VO. So they are alternate exterior.

Wait, but also, is there another? Wait, maybe the first option was misjudged. Wait, ∠RQT: is it outside? ∠RQT is at Q, between PR and VO. No, it's interior. ∠STQ: at T, between SU and VO. Interior. So not exterior.

∠STV and ∠STQ: adjacent, same vertex, not alternate.

∠UTQ and ∠STV: ∠UTQ is below SU, right of VO; ∠STV is above SU, left of VO. Wait, are they alternate exterior? Wait, ∠UTQ is exterior to SU (below the line), ∠STV is exterior to SU (above the line)? No, alternate exterior should be with respect to two lines. Wait, SU and PR are the two lines. So ∠STV is exterior to SU, ∠RQO is exterior to PR, and they are on alternate sides of the transversal. ∠UTQ is exterior to SU, but on the same side as ∠RQO? No, ∠UTQ is right of VO, ∠RQO is right of VO? Wait, no, VO is a transversal going from V (left) to O (right). So left of VO is the side towards V, right is towards O.

So ∠STV: left of VO, outside SU.

∠RQO: right of VO, outside PR.

So they are alternate (left and right) and exterior (outside the two lines SU and PR).

∠UTQ: right of VO, outside SU.

∠PQV (if it existed) would be left of VO, outside PR, but in the options, we have ∠RQO. So among the given options, ∠STV and ∠RQO are alternate exterior angles. Also, wait, maybe I made a mistake with ∠UTQ and ∠STV? No, ∠UTQ and ∠STV are adjacent angles (supplementary, forming a linear pair? Wait, ∠STV + ∠UTQ = 180°, so they are supplementary, not alternate exterior.

Wait, let's re - check the options again. The options are:

1. ∠RQT and ∠STQ

2. ∠STV and ∠STQ

3. ∠STV and ∠RQO

4. ∠UTQ and ∠STV

So the correct pair is ∠STV and ∠RQO. Wait, but maybe there is another? Wait, maybe the first option was wrong, and the third option is correct. Also, wait, maybe I misidentified the lines. The two lines are SU (S - T - U) and PR (P - Q - R). The transversal is VTQO (V - T - Q - O).

Exterior angles: For line SU, the exterior angles relative to transversal VTQO are ∠STV (above SU, left of VTQO) and ∠UTQ (below SU, right of VTQO). For line PR, the exterior angles relative to transversal VTQO are ∠PQV (above PR, left of VTQO) and ∠RQO (below PR, right of VTQO).

Alternate exterior angles would be ∠STV (exterior to SU, left) and ∠RQO (exterior to PR, right) – alternate sides (left and right) and exterior. Also, ∠UTQ (exterior to SU, right) and ∠PQV (exterior to PR, left) – but ∠PQV is not in the options. So among the given options, ∠STV and ∠RQO are alternate exterior angles. Also, wait, is ∠RQT and ∠STQ? No, they are interior angles (between the two lines). ∠STV and ∠STQ: same vertex, not alternate. ∠UTQ and ∠STV: supplementary, not alternate exterior. So the correct pair is ∠STV and ∠RQO.

Answer:

∠STV and ∠RQO