Question

select all the pairs of corresponding angles.
∠pqs and ∠mnq ∠onl and ∠rqn
∠rqs and ∠mnq ∠mnl and ∠rqn

Explanation:

Step1: Recall Corresponding Angles Definition

Corresponding angles are equal when two parallel lines are cut by a transversal, and they occupy the same relative position at each intersection.

Step2: Analyze Each Pair

- For $\angle PQS$ and $\angle MNQ$: Check their positions. $\angle PQS$ is at the lower intersection (with transversal $LS$ and line $PR$), $\angle MNQ$ is at the upper intersection (with transversal $LS$ and line $MO$). Their positions don't match corresponding angle positions.
- For $\angle ONL$ and $\angle RQN$: $\angle ONL$ is at the upper intersection (top - right of transversal on line $MO$), $\angle RQN$ is at the lower intersection (top - right of transversal on line $PR$). Same relative position, so they are corresponding angles.
- For $\angle RQS$ and $\angle MNQ$: $\angle RQS$ is at the lower intersection (bottom - right of transversal on line $PR$), $\angle MNQ$ is at the upper intersection (bottom - left of transversal on line $MO$). Positions don't match.
- For $\angle MNL$ and $\angle RQN$: $\angle MNL$ is at the upper intersection (top - left of transversal on line $MO$), $\angle RQN$ is at the lower intersection (top - right of transversal on line $PR$). Wait, no, re - check: $\angle MNL$ (top - left of transversal $LS$ on $MO$) and $\angle RQN$ (top - right of transversal $LS$ on $PR$) – no, wait, actually $\angle MNL$ and $\angle RQN$: Let's re - visualize. Line $MO$ and $PR$ are parallel (assumed as they are cut by transversal $LS$). $\angle MNL$ is above $MO$ and to the left of $LS$, $\angle RQN$ is above $PR$ and to the right of $LS$? No, wait, maybe I made a mistake. Wait, $\angle ONL$ and $\angle RQN$: $\angle ONL$ is on $MO$, right of $LS$, above $MO$; $\angle RQN$ is on $PR$, right of $LS$, above $PR$. So same relative position. Also, $\angle MNL$: on $MO$, left of $LS$, above $MO$; $\angle PQN$? Wait, no, the fourth pair: $\angle MNL$ and $\angle RQN$ – wait, $\angle MNL$ is at the upper intersection, left - top, $\angle RQN$ is at lower intersection, right - top. No, wait, maybe another way: Corresponding angles are in the same "corner" relative to the transversal and the parallel lines. So when transversal $LS$ cuts parallel lines $MO$ and $PR$:
- Angles above $MO$ and above $PR$, same side of transversal: $\angle ONL$ (above $MO$, right of $LS$) and $\angle RQN$ (above $PR$, right of $LS$) – corresponding.
- Angles above $MO$ and above $PR$, left of transversal: $\angle MNL$ (above $MO$, left of $LS$) and $\angle PQN$? But $\angle PQN$ is not an option. Wait, the option is $\angle MNL$ and $\angle RQN$ – no, maybe I messed up. Wait, the correct pairs are $\angle ONL$ and $\angle RQN$, and also $\angle MNL$ and $\angle PQS$? But $\angle PQS$ is not in the options. Wait, the given options: Let's re - list the options:
1. $\angle PQS$ and $\angle MNQ$
2. $\angle ONL$ and $\angle RQN$
3. $\angle RQS$ and $\angle MNQ$
4. $\angle MNL$ and $\angle RQN$

Wait, maybe I was wrong earlier. Let's take each pair:

- $\angle ONL$ and $\angle RQN$: Transversal $LS$, lines $MO\parallel PR$. $\angle ONL$ is on $MO$, exterior, right of $LS$; $\angle RQN$ is on $PR$, exterior, right of $LS$. Same relative position – corresponding angles.
- $\angle MNL$ and $\angle RQN$: No, $\angle MNL$ is on $MO$, interior, left of $LS$; $\angle RQN$ is on $PR$, exterior, right of $LS$. Not corresponding. Wait, no, $\angle MNL$: let's see the angle at $N$: $\angle MNL$ is between $MN$ (left part of $MO$) and $NL$ (transversal). $\angle RQN$ is between $RQ$ (right part of $PR$) and $QN$ (transversal). Wait, maybe the correct pairs are $\angle ONL$ and $\angle RQN$ and $\angle MNL$ and $\angle PQS$ (but $\angle PQS$ is not an option). Wait, the options given: the second pair ($\angle ONL$ and $\angle RQN$) and the fourth pair ($\angle MNL$ and $\angle RQN$)? No, I think I made a mistake. Wait, let's look at the diagram again. Line $MO$ (with points $M$, $N$, $O$: $M$ left, $O$ right) and line $PR$ (with points $P$, $Q$, $R$: $P$ left, $R$ right). Transversal $LS$ (with points $L$, $N$, $Q$, $S$: $L$ top, $S$ bottom).

- $\angle ONL$: at $N$, between $ON$ (right of $MO$) and $NL$ (transversal, going to $L$). So above $MO$, right of $LS$.
- $\angle RQN$: at $Q$, between $RQ$ (right of $PR$) and $QN$ (transversal, going to $N$). So above $PR$, right of $LS$. Same relative position – corresponding.

- $\angle MNL$: at $N$, between $MN$ (left of $MO$) and $NL$ (transversal). So above $MO$, left of $LS$.
- $\angle PQN$: at $Q$, between $PQ$ (left of $PR$) and $QN$ (transversal). But $\angle PQN$ is not an option. The fourth option is $\angle MNL$ and $\angle RQN$ – no, $\angle RQN$ is right of $LS$, $\angle MNL$ is left of $LS$. So that's not corresponding. Wait, maybe the fourth pair is $\angle MNL$ and $\angle PQS$? No, $\angle PQS$ is at $Q$, between $PQ$ (left of $PR$) and $QS$ (transversal, going to $S$). So below $PR$, left of $LS$. $\angle MNL$ is above $MO$, left of $LS$. Not corresponding.

Wait, maybe the correct pairs are $\angle ONL$ and $\angle RQN$ and $\angle MNL$ and $\angle PQS$ (but $\angle PQS$ is not an option). Wait, the options:

Wait, the problem says "select all pairs". Let's re - evaluate each option:

1. $\angle PQS$ and $\angle MNQ$: $\angle PQS$ is below $PR$, right of $LS$; $\angle MNQ$ is below $MO$, left of $LS$. Not corresponding.
2. $\angle ONL$ and $\angle RQN$: As above, same relative position – corresponding.
3. $\angle RQS$ and $\angle MNQ$: $\angle RQS$ is below $PR$, right of $LS$; $\angle MNQ$ is below $MO$, left of $LS$. Not corresponding.
4. $\angle MNL$ and $\angle RQN$: $\angle MNL$ is above $MO$, left of $LS$; $\angle RQN$ is above $PR$, right of $LS$. Wait, no, maybe I got the direction wrong. Maybe the lines are horizontal, $MO$ and $PR$ are horizontal, transversal $LS$ is slanting. So $\angle MNL$: at $N$, top - left of transversal on $MO$; $\angle RQN$: at $Q$, top - right of transversal on $PR$ – no, that's not corresponding. Wait, maybe the answer is $\angle ONL$ and $\angle RQN$ and $\angle MNL$ and $\angle RQN$? No, I think I made a mistake. Wait, maybe the fourth pair: $\angle MNL$ and $\angle RQN$ – no, let's check the definition again. Corresponding angles are in the same position relative to the parallel lines and the transversal. So if two parallel lines are cut by a transversal, corresponding angles are equal and are in the same "corner" (e.g., top - right, top - left, bottom - right, bottom - left) at each intersection.

So for line $MO$ (upper parallel) and $PR$ (lower parallel), cut by transversal $LS$:

- Top - right corner: $\angle ONL$ (on $MO$) and $\angle RQN$ (on $PR$) – corresponding.
- Top - left corner: $\angle MNL$ (on $MO$) and $\angle PQN$ (on $PR$) – but $\angle PQN$ is not an option. The fourth option is $\angle MNL$ and $\angle RQN$ – which is top - left on $MO$ and top - right on $PR$ – not same corner. So maybe only $\angle ONL$ and $\angle RQN$? But the fourth option is $\angle MNL$ and $\angle RQN$ – maybe I'm mis - visualizing. Wait, maybe the lines are $MO$ and $PR$ are parallel, transversal $LS$. $\angle MNL$ is adjacent to $\angle ONL$ (supplementary), and $\angle RQN$ is adjacent to $\angle PQN$ (supplementary). Wait, maybe the correct pairs are $\angle ONL$ and $\angle RQN$ and $\angle MNL$ and $\angle RQN$? No, that can't be. Wait, let's look at the options again. The user's options:

1. $\angle PQS$ and $\angle MNQ$
2. $\angle ONL$ and $\angle RQN$
3. $\angle RQS$ and $\angle MNQ$
4. $\angle MNL$ and $\angle RQN$

After re - analyzing, the pairs of corresponding angles are $\angle ONL$ and $\angle RQN$, and $\angle MNL$ and $\angle RQN$? No, I think the correct ones are $\angle ONL$ and $\angle RQN$ (second option) and $\angle MNL$ and $\angle RQN$ (fourth option)? Wait, no, maybe the fourth option is $\angle MNL$ and $\angle RQN$ – no, $\angle MNL$ is at $N$, $\angle RQN$ is at $Q$. Wait, maybe the answer is $\angle ONL$ and $\angle RQN$ (option 2) and $\angle MNL$ and $\angle RQN$ (option 4)? No, I'm confused. Wait, let's check with an example. If you have two parallel lines and a transversal, corresponding angles are like:

- Upper - right on first line and upper - right on second line.
- Upper - left on first line and upper - left on second line.
- Lower - right on first line and lower - right on second line.
- Lower - left on first line and lower - left on second line.

So $\angle ONL$: upper - right on $MO$ (since $O$ is right, $N$ is intersection, $L$ is top, so above $MO$, right of $LS$). $\angle RQN$: upper - right on $PR$ ( $R$ is right, $Q$ is intersection, $N$ is top - side of transversal, so above $PR$, right of $LS$). So they are corresponding.

$\angle MNL$: upper - left on $MO$ ( $M$ is left, $N$ is intersection, $L$ is top, so above $MO$, left of $LS$). $\angle PQN$: upper - left on $PR$ ( $P$ is left, $Q$ is intersection, $N$ is top - side of transversal, so above $PR$, left of $LS$). But $\angle PQN$ is not an option. The fourth option is $\angle MNL$ and $\angle RQN$ – $\angle RQN$ is upper - right on $PR$, so not same as upper - left on $MO$. So maybe the only correct pair is $\angle ONL$ and $\angle RQN$? But the problem says "select all", so maybe I missed. Wait, $\angle RQS$ and $\angle MNQ$: $\angle RQS$ is lower - right on $PR$ ( $R$ right, $Q$ intersection, $S$ bottom, so below $PR$, right of $LS$), $\angle MNQ$ is lower - left on $MO$ ( $M$ left, $N$ intersection, $Q$? No, $\angle MNQ$ is at $N$, below $MO$, left of $LS$). Not corresponding. $\angle PQS$ and $\angle MNQ$: $\angle PQS$ is below $PR$, left of $LS$; $\angle MNQ$ is below $MO$, left of $LS$. Oh! Wait, $\angle PQS$: at $Q$, below $PR$, left of $LS$ (since $P$ is left, $Q$ is intersection, $S$ is bottom, so angle between $PQ$ (left) and $QS$ (transversal down) – so below $PR$, left of $LS$. $\angle MNQ$: at $N$, below $MO$, left of $LS$ (between $MN$ (left) and $NQ$ (transversal down)). So same relative position (below the parallel lines, left of transversal). So $\angle PQS$ and $\angle MNQ$ are also corresponding angles! Oh, I missed that. So:

- $\angle PQS$ and $\angle MNQ$: below the parallel lines, left of transversal – corresponding.
- $\angle ONL$ and $\angle RQN$: above the parallel lines, right of transversal – corresponding.
- $\angle MNL$ and $\angle RQN$: No, $\angle MNL$ is above, left; $\angle RQN$ is above, right.
- $\angle RQS$ and $\angle MNQ$: $\angle RQS$ is below, right; $\angle MNQ$ is below, left – not corresponding.

Wait, so now I see: $\angle PQS$ (below $PR$, left of $LS$) and $\angle MNQ$ (below $MO$, left of $LS$) – corresponding. $\angle ONL$ (above $MO$, right of $LS$) and $\angle RQN$ (above $PR$, right of $LS$) – corresponding. And what about $\angle MNL$ (above $MO$, left of $LS$) and $\angle PQN$ (above $PR$, left of $LS$) – but $\angle PQN$ is not an option. The fourth option: $\angle MNL$ and $\angle RQN$ – no. So the correct pairs are $\angle PQS$ and $\angle MNQ$, $\angle ONL$ and $\angle RQN$. Wait, but the first option is $\angle PQS$ and $\angle MNQ$, second is $\angle ONL$ and $\angle RQN$. So those two are correct.

Step3: Confirm the Pairs

- $\angle PQS$ and $\angle MNQ$: Both are in the lower - left position relative to the transversal and their respective parallel lines ( $MO$ and $PR$), so corresponding.
- $\angle ONL$ and $\angle RQN$: Both are in the upper - right position relative to the transversal and their respective parallel lines, so corresponding.

Answer:

$\angle PQS$ and $\angle MNQ$, $\angle ONL$ and $\angle RQN$ (i.e., the first option: $\boldsymbol{\angle PQS}$ and $\boldsymbol{\angle MNQ}$, the second option: $\boldsymbol{\angle ONL}$ and $\boldsymbol{\angle RQN}$)