Question

qs and tv are parallel lines.
which angles are vertical angles?
∠vur and ∠srp ∠vur and ∠tuw
∠vur and ∠qrp ∠vur and ∠vuw

Explanation:

Step1: Recall vertical angles definition

Vertical angles are opposite angles formed by two intersecting lines, and they are equal.

Step2: Analyze each option

- Option 1: $\angle VUR$ and $\angle SRP$: $\angle SRP$ is at $R$, $\angle VUR$ at $U$. Not vertical (different intersections).
- Option 2: $\angle VUR$ and $\angle TUW$: Wait, no, let's re - check. Wait, the intersecting line is $PW$ (from $P$ to $W$). At intersection $U$, the angles formed with line $TV$: $\angle VUR$ and $\angle TUW$? Wait, no, wait the correct vertical angle for $\angle VUR$: when line $PW$ intersects $TV$ at $U$, the vertical angle of $\angle VUR$ should be... Wait, no, let's look at the other option: Wait, $\angle VUR$ and $\angle QRP$? Wait, no, let's re - examine the diagram. Wait, the two parallel lines are $QS$ and $TV$, and the transversal is $PW$. At point $R$ (on $QS$) and $U$ (on $TV$). Vertical angles are formed when two lines intersect. So at point $R$, the angles with transversal $PW$ and line $QS$; at point $U$, with transversal $PW$ and line $TV$. Now, $\angle VUR$: let's see the sides. $\angle VUR$ has sides $UV$ and $UR$. The vertical angle should be the angle opposite when $PW$ and $TV$ intersect. Wait, no, maybe I made a mistake. Wait, the correct pair: $\angle VUR$ and $\angle QRP$? Wait, no, let's check the options again. Wait, the first option: $\angle VUR$ and $\angle SRP$: $\angle SRP$ is at $R$, between $SR$ and $RP$. $\angle VUR$ is at $U$, between $UV$ and $UR$. Since $QS\parallel TV$, and $PW$ is transversal, $\angle SRP$ and $\angle VUR$: are they corresponding? No, vertical angles must be from the same intersection? Wait, no, vertical angles are from two intersecting lines. So the two intersecting lines are $PW$ and $QS$ (at $R$), and $PW$ and $TV$ (at $U$). Wait, no, $PW$ is a single line, so it intersects $QS$ at $R$ and $TV$ at $U$. So the angles at $R$ and $U$: $\angle QRP$ (at $R$, between $QR$ and $RP$) and $\angle VUR$ (at $U$, between $UV$ and $UR$): since $QS\parallel TV$, but vertical angles are from the same intersection? Wait, no, I think I messed up. Wait, the correct vertical angle for $\angle VUR$: when line $PW$ intersects $TV$ at $U$, the vertical angle of $\angle VUR$ is $\angle TUW$? No, wait, let's look at the last option? Wait, no, the correct answer is $\angle VUR$ and $\angle QRP$? Wait, no, let's start over.

Wait, vertical angles are formed by two intersecting lines. So when two lines intersect, they form two pairs of vertical angles. So the line $PW$ (from $P$ to $W$) intersects line $QS$ at $R$ and line $TV$ at $U$. So at intersection $R$ (between $QS$ and $PW$), the angles are: $\angle QRP$ (between $QR$ and $RP$) and $\angle SRW$ (opposite), and $\angle QRW$ and $\angle SRP$ (opposite). At intersection $U$ (between $TV$ and $PW$), the angles are: $\angle VUR$ (between $UV$ and $UR$) and $\angle TUW$ (between $TU$ and $UW$), and $\angle VUW$ and $\angle TU R$ (wait, no). Wait, now, $\angle QRP$ (at $R$) and $\angle VUR$ (at $U$): since $QS\parallel TV$, and $PW$ is transversal, $\angle QRP$ and $\angle VUR$: are they vertical angles? Wait, no, because they are at different intersections. Wait, I think I made a mistake in the initial analysis. Let's check the definition again: Vertical angles are equal and opposite angles formed by the intersection of two lines. So the two lines must intersect at a single point. So in this case, the only way is that the two intersecting lines are $PW$ and... Wait, $QS$ and $TV$ are parallel, not intersecting. So the intersecting lines are $PW$ and $QS$ (at $R$), and $PW$ and $TV$ (at $U$). So the vertical angles at $R$: $\angle QRP$ and $\angle SRW$; $\angle QRW$ and $\angle SRP$. At $U$: $\angle VUR$ and $\angle TUW$; $\angle VUW$ and $\angle TU R$. Wait, but the option has $\angle VUR$ and $\angle QRP$. Wait, maybe the diagram is such that $\angle QRP$ and $\angle VUR$ are vertical angles? Wait, no, unless $QS$ and $TV$ are the same line, which they are not (they are parallel). Wait, I think I misread the options. Wait, the third option is $\angle VUR$ and $\angle QRP$. Let's check the positions: $\angle QRP$ is at $R$, between $QR$ (part of $QS$) and $RP$ (part of $PW$). $\angle VUR$ is at $U$, between $UV$ (part of $TV$) and $UR$ (part of $PW$). Since $QS\parallel TV$, and $PW$ is a transversal, $\angle QRP$ and $\angle VUR$: are they alternate - interior angles? No, vertical angles. Wait, maybe the problem considers that the two intersections (at $R$ and $U$) are such that $\angle QRP$ and $\angle VUR$ are vertical angles? But according to the strict definition, vertical angles must be at the same intersection. So I must have made a mistake. Wait, let's look at the options again:

The options are:

1. $\angle VUR$ and $\angle SRP$

2. $\angle VUR$ and $\angle TUW$

3. $\angle VUR$ and $\angle QRP$

4. $\angle VUR$ and $\angle VUW$

Now, $\angle VUW$: at $U$, between $UV$ and $UW$. $\angle VUR$ is between $UV$ and $UR$. So they are adjacent, not vertical. $\angle TUW$: between $TU$ and $UW$. $\angle VUR$: between $UV$ and $UR$. $TU$ is opposite to $UV$ (since $TV$ is a straight line, $TU$ and $UV$ are opposite rays), and $UR$ and $UW$ are opposite rays (since $PW$ is a straight line, $UR$ and $UW$ are opposite rays). So $\angle VUR$ (between $UV$ and $UR$) and $\angle TUW$ (between $TU$ and $UW$): these are vertical angles! Because when two lines ($TV$ and $PW$) intersect at $U$, $\angle VUR$ and $\angle TUW$ are opposite angles. Wait, earlier I thought wrong. Let's confirm: when two lines intersect, the vertical angles are the pairs of opposite angles. So line $TV$ (horizontal) and line $PW$ (slanted) intersect at $U$. So the four angles at $U$: $\angle VUR$, $\angle VUW$, $\angle TUW$, $\angle TUR$. So $\angle VUR$ and $\angle TUW$: $\angle VUR$ has sides $UV$ (right along $TV$) and $UR$ (up along $PW$), $\angle TUW$ has sides $TU$ (left along $TV$) and $UW$ (down along $PW$). So they are vertical angles. Wait, but in the option, it's $\angle VUR$ and $\angle TUW$? Wait, no, the second option is $\angle VUR$ and $\angle TUW$? Wait, the user's options: let's re - check the original problem's options:

Wait, the user wrote:

$\angle VUR$ and $\angle SRP$

$\angle VUR$ and $\angle TUW$

$\angle VUR$ and $\angle QRP$

$\angle VUR$ and $\angle VUW$

Wait, maybe I misread the second option. Let's assume that the correct vertical angle is $\angle VUR$ and $\angle QRP$? No, that can't be. Wait, no, $\angle QRP$ is at $R$, on line $QS$. $\angle VUR$ is at $U$, on line $TV$. Since $QS\parallel TV$, and $PW$ is transversal, $\angle QRP$ and $\angle VUR$: are they corresponding angles? No, vertical angles must be from the same intersection. So the only possible vertical angles at the same intersection (at $U$) are $\angle VUR$ and $\angle TUW$? Wait, but $TU$ and $UV$ are a straight line (since $TV$ is a straight line), and $UR$ and $UW$ are a straight line (since $PW$ is a straight line). So when $TV$ and $PW$ intersect at $U$, $\angle VUR$ (between $UV$ and $UR$) and $\angle TUW$ (between $TU$ and $UW$) are vertical angles. But wait, in the option, is it $\angle VUR$ and $\angle TUW$? Wait, the user's second option is $\angle VUR$ and $\angle TUW$? Wait, no, maybe the correct answer is $\angle VUR$ and $\angle QRP$? I think I made a mistake in the ray directions. Let's look at the diagram again: $QS$ is from $Q$ (left) to $S$ (right), $TV$ is from $T$ (left) to $V$ (right). The transversal is from $P$ (top) to $W$ (bottom). At point $R$ (on $QS$), the angle $\angle QRP$: $Q$ is left, $R$ is the intersection, $P$ is top. At point $U$ (on $TV$), $\angle VUR$: $V$ is right, $U$ is intersection, $R$ is... Wait, no, $UR$ is part of the transversal going down to $W$. So $\angle QRP$ (at $R$: left - top) and $\angle VUR$ (at $U$: right - top) – since $QS\parallel TV$, these two angles are equal (corresponding angles), but are they vertical? No. Wait, the correct vertical angle for $\angle VUR$ is $\angle TUW$? Wait, $TU$ is left, $UW$ is bottom. $\angle VUR$ is right, $UR$ is top. So they are vertical. Yes, because when two lines intersect, vertical angles are opposite. So line $TV$ (horizontal) and line $PW$ (slanted) intersect at $U$. So $\angle VUR$ (between $UV$ (right) and $UR$ (top)) and $\angle TUW$ (between $TU$ (left) and $UW$ (bottom)) are vertical angles. But wait, in the option, is the second option $\angle VUR$ and $\angle TUW$? The user wrote: $\angle VUR$ and $\angle TUW$? Wait, no, the user's second option is $\angle VUR$ and $\angle TUW$? Wait, maybe I made a mistake in the first analysis. Let's confirm with the definition: vertical angles are equal and opposite when two lines cross. So at intersection $U$, the two lines are $TV$ (straight line: $T - U - V$) and $PW$ (straight line: $P - U - W$). So the four angles: $\angle VUR$ ( $V - U - R$), $\angle VUW$ ( $V - U - W$), $\angle TUW$ ( $T - U - W$), $\angle TUR$ ( $T - U - R$). So $\angle VUR$ and $\angle TUW$ are vertical angles (opposite each other), and $\angle VUW$ and $\angle TUR$ are vertical angles. So the correct option is $\angle VUR$ and $\angle TUW$? But wait, the user's options: let's check again. Wait, the third option is $\angle VUR$ and $\angle QRP$. Wait, maybe the diagram is different. Wait, maybe $\angle QRP$ and $\angle VUR$ are vertical angles because the transversal and the two parallel lines form a sort of "X" at two points, and these angles are opposite. I think I was wrong earlier. Let's use the definition strictly: vertical angles are formed by two intersecting lines, so they must share the same vertex. So $\angle VUR$ has vertex $U$, so its vertical angle must also have vertex $U$. So among the options, only $\angle VUR$ and $\angle TUW$ (vertex $U$), $\angle VUR$ and $\angle VUW$ (vertex $U$) are at vertex $U$. $\angle VUW$ is adjacent to $\angle VUR$ (they share a side $UV$), so they are supplementary, not vertical. $\angle TUW$: shares the vertex $U$, and is opposite to $\angle VUR$ (since $TV$ and $PW$ intersect at $U$). So $\angle VUR$ and $\angle TUW$ are vertical angles. Wait, but in the option, is that the case? The user's second option is $\angle VUR$ and $\angle TUW$? If so, that's the answer. But wait, maybe I misread the option. Wait, the user wrote: $\angle VUR$ and $\angle TUW$? Let's assume that's the case. So the correct answer is the second option: $\angle VUR$ and $\angle TUW$. Wait, no, wait the original problem's options: maybe I made a mistake. Wait, let's look at the first option: $\angle VUR$ and $\angle SRP$. $\angle SRP$ is at $R$, between $SR$ (right) and $RP$ (top). $\angle VUR$ is at $U$, between $UV$ (right) and $UR$ (top). Since $QS\parallel TV$, these two angles are equal (corresponding angles), but not vertical. The third option: $\angle VUR$ and $\angle QRP$. $\angle QRP$ is at $R$, between $QR$ (left) and $RP$ (top). $\angle VUR$ is at $U$, between $UV$ (right) and $UR$ (top). These are equal (alternate - interior angles? No, corresponding). The fourth option: $\angle VUR$ and $\angle VUW$: adjacent, supplementary. So the correct answer is $\angle VUR$ and $\angle TUW$ (second option).

Answer:

$\angle VUR$ and $\angle TUW$ (the option with $\angle VUR$ and $\angle TUW$)