Question

select all the pairs of alternate exterior angles.
∠tus and ∠vus ∠yxz and ∠tus
∠vus and ∠wxz ∠vus and ∠yxu

Explanation:

Step1: Recall Alternate Exterior Angles

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

Step2: Analyze Each Pair

- **∠TUS and ∠VUS**: These are adjacent angles, not alternate exterior.
- **∠YXZ and ∠TUS**: ∠YXZ is outside the two lines (YW and TV) and on one side of transversal ZS, ∠TUS is outside on the opposite side. Wait, no, recheck. Wait, YW and TV are parallel? Assuming the two vertical lines are parallel. ∠YXZ: let's see positions. Wait, maybe better to check the other pairs.
- **∠VUS and ∠WXZ**: ∠VUS is outside (below TV) and on right of transversal, ∠WXZ is outside (above YW) and on left of transversal. Wait, no. Wait, let's label the lines: YW (with W top, Y bottom) and TV (T top, V bottom) are parallel, transversal ZS (Z left, S right). Exterior angles are outside YW and TV. So ∠WXZ: at X, above YW, left of transversal. ∠VUS: at U, below TV, right of transversal. Wait, maybe another pair. Wait, the correct pair: ∠VUS and ∠WXZ? Wait, no, let's check the last option: ∠VUS and ∠YXU? No. Wait, maybe I made a mistake. Wait, the correct alternate exterior angles should be ∠YXZ and ∠TUS? No, wait, let's re-express. Wait, the two parallel lines are YW (X is the intersection with transversal) and TV (U is the intersection). Transversal ZS. Exterior angles: for line YW, exterior angles are above YW (∠WXZ) and below YW (∠YXZ? Wait, no, YXZ is at X, left of transversal, below YW? Wait, maybe the correct pair is ∠VUS and ∠WXZ? Wait, no, let's check the options again. Wait, the options are:

1. ∠TUS and ∠VUS: adjacent, linear pair? No, alternate exterior.
2. ∠YXZ and ∠TUS: ∠YXZ is at X, left of transversal, below YW (exterior? YW is between X and Y, so below YW is exterior? Wait, YW is a vertical line, transversal ZS crosses it at X. So the two regions: between YW and TV (interior) and outside (exterior: above YW and below TV, or below YW and above TV? Wait, maybe I messed up. Let's start over.

Alternate exterior angles: two angles, outside the two parallel lines, on opposite sides of the transversal.

So lines: YW (W---X---Y) and TV (T---U---V), parallel. Transversal Z---X---U---S.

Exterior angles:

- Above YW: ∠WXZ (at X, above YW, left of transversal)
- Below TV: ∠VUS (at U, below TV, right of transversal)
- Below YW: ∠YXZ (at X, below YW, left of transversal)
- Above TV: ∠TUS (at U, above TV, right of transversal)

So alternate exterior angles would be ∠WXZ (above YW, left) and ∠VUS (below TV, right) – no, opposite sides? Wait, transversal is ZS, so left and right. ∠WXZ is left of transversal, ∠VUS is right of transversal. They are outside the two lines (YW and TV) – yes, ∠WXZ is outside (above YW, which is one line, TV is the other, so above YW is outside the region between YW and TV), and ∠VUS is outside (below TV, outside the region between YW and TV). So ∠VUS and ∠WXZ: are they alternate exterior? Wait, no, alternate exterior should be on opposite sides of the transversal and outside the two lines. Wait, maybe ∠YXZ (below YW, left) and ∠TUS (above TV, right). Let's check: ∠YXZ is below YW (outside the middle region), left of transversal. ∠TUS is above TV (outside the middle region), right of transversal. So they are on opposite sides of transversal and outside the two lines. So ∠YXZ and ∠TUS? Wait, but the option is ∠YXZ and ∠TUS? Wait, the second option is ∠YXZ and ∠TUS. Wait, but also, ∠VUS and ∠WXZ: ∠VUS is below TV (outside), right of transversal. ∠WXZ is above YW (outside), left of transversal. So they are alternate exterior. Wait, maybe both? But the problem says "select all". Wait, maybe I made a mistake. Wait, let's check the diagram again. The two vertical lines: YW (X is the intersection) and TV (U is the intersection). Transversal ZS (Z to S). So the angles:

- At X: angles above YW (∠WXZ) and below YW (∠YXZ) – these are exterior? Wait, YW is a line, so the two sides of YW are interior (between YW and TV) and exterior (outside YW and TV). So above YW (towards W) is exterior, below YW (towards Y) is exterior? No, the interior is between YW and TV, so exterior is above YW (away from TV) and below TV (away from YW). So ∠WXZ: above YW, left of transversal (exterior). ∠VUS: below TV, right of transversal (exterior). These are on opposite sides of transversal (left and right) and outside the two lines (YW and TV). So ∠VUS and ∠WXZ are alternate exterior. Also, ∠YXZ: below YW, left of transversal (exterior) and ∠TUS: above TV, right of transversal (exterior). So ∠YXZ and ∠TUS are also alternate exterior? Wait, but the options given:

Options:

1. ∠TUS and ∠VUS: adjacent, not alternate exterior.
2. ∠YXZ and ∠TUS: yes, alternate exterior (below YW, left; above TV, right – outside, opposite sides).
3. ∠VUS and ∠WXZ: yes, alternate exterior (below TV, right; above YW, left – outside, opposite sides).
4. ∠VUS and ∠YXU: ∠YXU is at X, below YW, right of transversal – interior? No.

Wait, but maybe the diagram has YW and TV parallel, so the correct alternate exterior angles are ∠YXZ and ∠TUS, and ∠VUS and ∠WXZ? But the options given: let's check the original options again. The user provided:

Options:

- ∠TUS and ∠VUS

- ∠YXZ and ∠TUS

- ∠VUS and ∠WXZ

- ∠VUS and ∠YXU

Wait, maybe I misread. Let's re-express:

Wait, the correct pairs are ∠YXZ and ∠TUS? No, wait, ∠YXZ: at X, left of transversal, below YW (exterior). ∠TUS: at U, above TV, right of transversal (exterior). So they are alternate exterior. And ∠VUS: at U, below TV, right of transversal (exterior). ∠WXZ: at X, above YW, left of transversal (exterior). So these two are also alternate exterior. But maybe the problem considers only one? Wait, no, the problem says "select all". Wait, maybe the diagram is such that YW and TV are parallel, so the alternate exterior angles are ∠VUS and ∠WXZ, and ∠YXZ and ∠TUS? But let's check the options. Wait, the user's options:

First option: ∠TUS and ∠VUS – adjacent, not alternate.

Second: ∠YXZ and ∠TUS – let's see, ∠YXZ is at X, left of transversal, below YW. ∠TUS is at U, above TV, right of transversal. So outside the two lines (YW and TV) and opposite sides of transversal. So yes, alternate exterior.

Third: ∠VUS and ∠WXZ – ∠VUS is at U, below TV, right of transversal. ∠WXZ is at X, above YW, left of transversal. Outside the two lines, opposite sides of transversal. So yes, alternate exterior.

Fourth: ∠VUS and ∠YXU – ∠YXU is at X, below YW, right of transversal – interior (between YW and TV), so no.

So the correct pairs are ∠YXZ and ∠TUS, and ∠VUS and ∠WXZ. But wait, maybe the diagram is different. Wait, maybe the two vertical lines are YW (X) and TV (U), transversal ZS. So ∠WXZ: above YW, left of transversal (exterior). ∠VUS: below TV, right of transversal (exterior). These are alternate exterior. ∠YXZ: below YW, left of transversal (exterior). ∠TUS: above TV, right of transversal (exterior). These are also alternate exterior. But the options given: the second option is ∠YXZ and ∠TUS, third is ∠VUS and ∠WXZ. So both are correct? But maybe the problem has only one correct? Wait, maybe I made a mistake. Let's check the definition again: alternate exterior angles are two angles that lie outside the two lines, on opposite sides of the transversal, and are congruent if lines are parallel. So in the diagram, assuming YW || TV, then:

- ∠WXZ (above YW, left) and ∠VUS (below TV, right) – alternate exterior.

- ∠YXZ (below YW, left) and ∠TUS (above TV, right) – alternate exterior.

So both pairs. But the options given: the second option is ∠YXZ and ∠TUS, third is ∠VUS and ∠WXZ. So both are correct? Wait, but maybe the diagram is drawn such that ∠VUS and ∠WXZ are alternate exterior, and ∠YXZ and ∠TUS are also. But let's check the options again. The user's options:

1. ∠TUS and ∠VUS: no.

2. ∠YXZ and ∠TUS: yes.

3. ∠VUS and ∠WXZ: yes.

4. ∠VUS and ∠YXU: no.

So the correct answers are ∠YXZ and ∠TUS, and ∠VUS and ∠WXZ. But maybe the problem expects one? Wait, no, the problem says "select all". Wait, maybe I misread the diagram. Let's assume that the two vertical lines are parallel, so the alternate exterior angles are ∠VUS and ∠WXZ, and ∠YXZ and ∠TUS. But let's check the labels again. ∠YXZ: at X, with Z, X, so ∠YXZ is angle at X between YX and XZ. ∠TUS: at U, between TU and US. ∠VUS: at U, between VU and US. ∠WXZ: at X, between WX and XZ. So ∠WXZ (WX is up, XZ is left) and ∠VUS (VU is down, US is right) – these are outside the two lines (YW and TV) and on opposite sides of transversal (left and right). So yes, alternate exterior. ∠YXZ (YX is down, XZ is left) and ∠TUS (TU is up, US is right) – outside the two lines, opposite sides of transversal. So yes, alternate exterior. So both pairs. But the options given: the second and third options. So the correct answers are ∠YXZ and ∠TUS, and ∠VUS and ∠WXZ. But maybe the problem has a typo, or I made a mistake. Wait, maybe the correct pair is ∠VUS and ∠WXZ, and ∠YXZ and ∠TUS. But let's go with the options. So the correct pairs are:

- ∠YXZ and ∠TUS

- ∠VUS and ∠WXZ

But let's check the original problem again. The user's diagram: Z---X---U---S (transversal), YW (W up, Y down) through X, TV (T up, V down) through U. So angles:

- At X: ∠WXZ (WX up, XZ left) – exterior (above YW, left of transversal)

- At U: ∠VUS (VU down, US right) – exterior (below TV, right of transversal) – alternate exterior.

- At X: ∠YXZ (YX down, XZ left) – exterior (below YW, left of transversal)

- At U: ∠TUS (TU up, US right) – exterior (above TV, right of transversal) – alternate exterior.

So both pairs. So the correct options are the second (∠YXZ and ∠TUS) and third (∠VUS and ∠WXZ) options.

Answer:

The pairs of alternate exterior angles are $\angle YXZ$ and $\angle TUS$, and $\angle VUS$ and $\angle WXZ$. So the correct options are:

• $\angle YXZ$ and $\angle TUS$

• $\angle VUS$ and $\angle WXZ$