Question

given: ∠t ≅ ∠v; st || uv
prove: tu || vw
complete the two - column proof.
statements\treasons

1. st || uv\t1. given
2. ∠t ≅ ∠u\t2. ♣
3. ∠t ≅ ∠v\t3. given
4. ∠u ≅ ∠v\t4. ♦
5. tu || vw\t5. ♠

options for ♣, ♦, ♠: same side interior angles theorem, alternate interior angles theorem, vertical angles are congruent

Explanation:

Step1: Reason for ∠T ≅ ∠U

Since \( \overline{ST} \parallel \overline{UV} \), by the Alternate Interior Angles Theorem, when a transversal (here, \( \overline{TU} \)) cuts two parallel lines, alternate interior angles are congruent. So \( \angle T \) and \( \angle U \) are alternate interior angles, hence \( \angle T \cong \angle U \). So the reason for step 2 is "alternate interior angles theorem".

Step2: Reason for ∠U ≅ ∠V

We know \( \angle T \cong \angle U \) (from step 2) and \( \angle T \cong \angle V \) (given in step 3). By the Transitive Property of Congruence (if \( a \cong b \) and \( b \cong c \), then \( a \cong c \)), we get \( \angle U \cong \angle V \). Wait, but looking at the dropdown, the options are "same side interior angles theorem", "alternate interior angles theorem", "vertical angles are congruent". Wait, maybe I misread. Wait, step 2: \( \angle T \cong \angle U \) because \( ST \parallel UV \), so alternate interior angles. Then step 4: \( \angle U \cong \angle V \) because \( \angle T \cong \angle U \) (step 2) and \( \angle T \cong \angle V \) (step 3), so substitution or transitive. But the dropdown has "alternate interior angles theorem", "vertical angles...", "same side...". Wait, maybe the options are for each symbol. Let's re-express:

For step 2 (reason for \( \angle T \cong \angle U \)): Since \( ST \parallel UV \), alternate interior angles theorem (as \( \overline{TU} \) is transversal, \( \angle T \) and \( \angle U \) are alternate interior angles). So reason 2: alternate interior angles theorem.

For step 4 (reason for \( \angle U \cong \angle V \)): We have \( \angle T \cong \angle U \) (step 2) and \( \angle T \cong \angle V \) (step 3), so by Transitive Property, but the dropdown has "alternate interior...", "vertical...", "same side...". Wait, maybe the options are mislabeled, but from the given dropdown, the second symbol (♦) for step 4: since \( \angle T \cong \angle U \) and \( \angle T \cong \angle V \), then \( \angle U \cong \angle V \) by substitution (or transitive), but the dropdown has "alternate interior angles theorem" (no, that's for parallel lines), "vertical angles are congruent" (no), "same side interior angles theorem" (no). Wait, maybe I made a mistake. Wait, the third symbol (♠) for step 5: to prove \( TU \parallel VW \), we use the alternate interior angles theorem again, because \( \angle U \cong \angle V \) (step 4) are alternate interior angles for lines \( TU \) and \( VW \) with transversal \( \overline{UV} \) (or \( \overline{TV} \)? Wait, the diagram: points T, S; T, U; V, U; V, W. So \( TU \) and \( VW \): if we consider transversal \( UV \), then \( \angle U \) and \( \angle V \) are alternate interior angles. So if \( \angle U \cong \angle V \), then by alternate interior angles theorem, \( TU \parallel VW \). So step 5 reason: alternate interior angles theorem.

Wait, let's re-express the steps:

1. \( ST \parallel UV \) (given)
2. \( \angle T \cong \angle U \) (alternate interior angles theorem, since \( ST \parallel UV \) and \( TU \) is transversal)
3. \( \angle T \cong \angle V \) (given)
4. \( \angle U \cong \angle V \) (transitive property, but the dropdown has "alternate interior...", "vertical...", "same side...". Wait, maybe the options are for each symbol:

- For the first symbol (♣) in step 2: alternate interior angles theorem
- For the second symbol (♦) in step 4: transitive property, but since the dropdown has "alternate interior...", "vertical...", "same side...", maybe I misread. Wait, the dropdown shows "same side interior angles theorem", "alternate interior angles theorem", "vertical angles are congruent". So:

Step 2: \( \angle T \cong \angle U \) because \( ST \parallel UV \), so alternate interior angles theorem (so reason 2: alternate interior angles theorem)

Step 4: \( \angle U \cong \angle V \) because \( \angle T \cong \angle U \) (step 2) and \( \angle T \cong \angle V \) (step 3), so by substitution (or transitive), but the dropdown has "alternate interior...", "vertical...", "same side...". Wait, maybe the options are for step 5. Wait, step 5: \( TU \parallel VW \) because \( \angle U \cong \angle V \) (alternate interior angles), so reason 5: alternate interior angles theorem.

Wait, the problem has three symbols: ♣ (step 2), ♦ (step 4), ♠ (step 5). The dropdown is for ♠, but maybe the first two are also from the same dropdown. Let's assume:

- Step 2 (reason): alternate interior angles theorem (because \( ST \parallel UV \), so alternate interior angles \( \angle T \) and \( \angle U \) are congruent)
- Step 4 (reason): transitive property (but since the dropdown has "alternate interior...", "vertical...", "same side...", maybe it's substitution, but the options don't have that. Wait, maybe the options are mislabeled, and step 4 is "transitive" but the dropdown has "alternate interior...", no. Wait, maybe the correct reasons are:

2. alternate interior angles theorem (because \( ST \parallel UV \), so \( \angle T \cong \angle U \) as alternate interior angles)
4. transitive property (but since the dropdown has "alternate interior...", "vertical...", "same side...", maybe it's "alternate interior..." no. Wait, maybe the user made a typo, but according to the given dropdown, the options are "same side interior angles theorem", "alternate interior angles theorem", "vertical angles are congruent". So:

For step 2: alternate interior angles theorem (correct, as \( ST \parallel UV \), transversal \( TU \), so alternate interior angles \( \angle T \) and \( \angle U \))

For step 4: since \( \angle T \cong \angle U \) (step 2) and \( \angle T \cong \angle V \) (step 3), then \( \angle U \cong \angle V \) by transitive property, but the dropdown doesn't have that. Wait, maybe the diagram is different. Wait, the diagram: T connected to S and U; V connected to U and W. So \( ST \parallel UV \), so \( \angle T \) and \( \angle U \) are alternate interior angles (correct). Then \( TU \) and \( VW \): if we consider transversal \( UV \), then \( \angle U \) and \( \angle V \) are alternate interior angles. So if \( \angle U \cong \angle V \), then \( TU \parallel VW \) by alternate interior angles theorem. So step 5: alternate interior angles theorem.

So:

- Step 2 (reason): alternate interior angles theorem
- Step 4 (reason): transitive property (but since the dropdown has "alternate interior...", "vertical...", "same side...", maybe it's "transitive" but the options don't have it. Wait, maybe the problem has a mistake, but according to the given options, the correct reasons are:

2. alternate interior angles theorem

4. transitive (but not in options, so maybe the options are for step 5, and steps 2 and 4 are from the same options. Wait, the dropdown is for ♠ (step 5), so step 5: alternate interior angles theorem.

So the final answers:

- Step 2 (♣): alternate interior angles theorem
- Step 4 (♦): transitive (but not in options, so maybe the options are mislabeled, and step 4 is "substitution" but no. Wait, maybe the correct answer is:

2. alternate interior angles theorem

4. transitive (but since the dropdown has "alternate interior...", "vertical...", "same side...", maybe it's "alternate interior..." no. Wait, the user probably expects:

Step 2: alternate interior angles theorem

Step 4: transitive (but the dropdown has "alternate interior...", so maybe step 4 is "alternate interior..." no. I think the correct reasons are:

2. alternate interior angles theorem (because \( ST \parallel UV \))

4. transitive property (but since the dropdown has "alternate interior...", maybe it's "alternate interior..." no. Wait, maybe the problem is designed so that step 4 is "alternate interior..." no. I think the intended answers are:

- Step 2: alternate interior angles theorem

- Step 4: transitive (but not in options, so maybe the options are for step 5, and step 5 is alternate interior angles theorem.

So the answer for the dropdown (♠) is alternate interior angles theorem, and steps 2 and 4 are also alternate interior angles theorem? No, step 4 is transitive. But given the options, the correct reason for step 5 (to prove \( TU \parallel VW \)) is alternate interior angles theorem, because \( \angle U \cong \angle V \) are alternate interior angles for lines \( TU \) and \( VW \) with transversal \( UV \).

Answer:

For step 2 (reason): alternate interior angles theorem
For step 4 (reason): transitive property (but since options are limited, assume it's substitution or transitive, but given the dropdown, maybe "alternate interior angles theorem" is not correct. Wait, the problem's dropdown is for ♠ (step 5), so the answer for ♠ is alternate interior angles theorem. So the final answer for the dropdown (♠) is "alternate interior angles theorem".