Question

complete the proof that $\angle q \cong \angle s$.
(image of a geometric figure with triangles and segments marked congruent)

	statement	reason
2	$\overline{qw} \cong \overline{st}$	given
3	$\overline{ru} \cong \overline{rv}$	given
4	$\overline{vw} \cong \overline{tu}$	given
5	$qu = qr + ru$	additive property of length
6	$sv = rs + rv$	additive property of length
7	$qu = rs + rv$	substitution
8	$qu = sv$	transitive property of equality
9	$uw = vw + uv$	additive property of length
10	$tv = tu + uv$	additive property of length
11	$uw = tu + uv$	substitution
12	$tv = uw$	transitive property of equality
13

(\add line\ button at the bottom)

Explanation:

Step1: Identify Triangle Congruence

We have \( \overline{QW} \cong \overline{ST} \) (given), \( QU = SV \) (from step 8), and \( TV = UW \) (from step 12). So, by SSS (Side - Side - Side) congruence criterion, \( \triangle QWV \cong \triangle STU \)? Wait, no, let's check the sides. Wait, actually, we have \( QU = SV \), \( QW = ST \), and \( UW = TV \) (since \( TV = UW \)). So, \( \triangle QWU \cong \triangle STV \) by SSS? Wait, no, let's re - express. The sides of \( \triangle Q \) (wait, the triangles are \( \triangle QUV \) and \( \triangle SVR \)? Wait, no, looking at the segments: \( QU = SV \), \( QW = ST \), and \( UW = TV \) (since \( TV = UW \)). So, \( \triangle QWU \cong \triangle STV \) by SSS. Then, corresponding angles \( \angle Q \) and \( \angle S \) are congruent.

Step2: State Triangle Congruence and Angle Congruence

Statement 13: \( \triangle QWU \cong \triangle STV \) (by SSS, since \( QW = ST \), \( QU = SV \), \( UW = TV \))
Reason: SSS Congruence Criterion (Side - Side - Side: if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent)
Then, since corresponding parts of congruent triangles are congruent (CPCTC), \( \angle Q \cong \angle S \). Wait, maybe a better way: after proving the triangles congruent, the corresponding angles are congruent.

Wait, let's re - check the steps. We have:
- \( QW\cong ST \) (given, step 2)
- \( QU = SV \) (step 8)
- \( UW = TV \) (step 12)

So, in \( \triangle QWU \) and \( \triangle STV \):
- \( QW = ST \) (given)
- \( QU = SV \) (proven)
- \( UW = TV \) (proven)

So, \( \triangle QWU\cong\triangle STV \) by SSS. Then, by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \( \angle Q\cong\angle S \).

So, Statement 13: \( \triangle QWU \cong \triangle STV \) (SSS)
Reason: SSS Congruence (since \( QW = ST \), \( QU = SV \), \( UW = TV \))
Then, the next statement (but we need to get \( \angle Q\cong\angle S \)). Wait, maybe the triangles are \( \triangle Q \) (wait, the correct triangles: let's see the labels. The segments: \( RS\cong QR \), \( RU\cong RV \), \( QW\cong ST \). So, the triangles are \( \triangle QUR \) and \( \triangle SVR \)? No, better to use the SSS on \( \triangle Q \) (the triangle with vertices \( Q \), \( U \), \( W \)) and \( \triangle S \), \( T \), \( V \)).

So, the 13th statement should be \( \triangle QWU \cong \triangle STV \) (by SSS, because \( QW = ST \), \( QU = SV \), \( UW = TV \)) and then the reason for \( \angle Q\cong\angle S \) is CPCTC. But since we need to complete the proof, let's structure it:

Statement 13: \( \triangle QWU \cong \triangle STV \)
Reason: SSS (Side - Side - Side) Congruence Postulate (because \( QW\cong ST \), \( QU = SV \), \( UW = TV \))
Then, Statement 14 (but the problem may just need the triangle congruence and then angle congruence. Wait, maybe I made a mistake. Let's start over.

We have:
1. \( RS\cong QR \) (given)
2. \( QW\cong ST \) (given)
3. \( RU\cong RV \) (given)
4. \( VW\cong TU \) (given)
5. \( QU = QR + RU \) (Additive Property)
6. \( SV = RS + RV \) (Additive Property)
7. \( QU = RS + RV \) (Substitution, since \( QR = RS \) from \( RS\cong QR \) and \( RU = RV \) from \( RU\cong RV \))
8. \( QU = SV \) (Transitive, since \( SV = RS + RV \) and \( QU = RS + RV \))
9. \( UW = VW + UV \) (Additive Property)
10. \( TV = TU + UV \) (Additive Property)
11. \( UW = TU + UV \) (Substitution, since \( VW = TU \) from \( VW\cong TU \))
12. \( TV = UW \) (Transitive, since \( TV = TU + UV \) and \( UW = TU + UV \))

Now, in \( \triangle QWU \) and \( \triangle STV \):
- \( QW\cong ST \) (given, step 2)
- \( QU\cong SV \) (from step 8, \( QU = SV \) implies \( \overline{QU}\cong\overline{SV} \))
- \( UW\cong TV \) (from step 12, \( UW = TV \) implies \( \overline{UW}\cong\overline{TV} \))

So, by SSS congruence, \( \triangle QWU\cong\triangle STV \)

Then, by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \( \angle Q\cong\angle S \)

So, the 13th statement can be \( \triangle QWU \cong \triangle STV \) with reason SSS, and then the 14th statement (but the problem may have only up to 13? Wait, the table has up to 13. Wait, maybe the triangles are \( \triangle Q \) (the triangle with sides \( QW \), \( QU \), \( UW \)) and \( \triangle S \) (triangle with sides \( ST \), \( SV \), \( TV \)). So, statement 13: \( \triangle QWU \cong \triangle STV \) (SSS), reason: SSS (since \( QW = ST \), \( QU = SV \), \( UW = TV \))

Then, the next step (if needed) would be \( \angle Q\cong\angle S \) by CPCTC. But since the problem is to prove \( \angle Q\cong\angle S \), after proving the triangles congruent, we can say that \( \angle Q\cong\angle S \) because corresponding parts of congruent triangles are congruent.

Answer:

Statement 13: \( \triangle QWU \cong \triangle STV \)
Reason: SSS (Side - Side - Side) Congruence Criterion (since \( QW\cong ST \), \( QU = SV \), \( UW = TV \))
Then, by CPCTC, \( \angle Q\cong\angle S \) (but if we just need the statement for \( \angle Q\cong\angle S \), the statement would be \( \angle Q\cong\angle S \) and reason CPCTC, with the prior statement being the triangle congruence. So, if we assume that statement 13 is to prove the triangles congruent, then statement 14 (or maybe the last statement) is \( \angle Q\cong\angle S \) by CPCTC. But based on the given table, after step 12, we need to complete the proof. So, the key is that after showing the three sides are congruent, the triangles are congruent, and then the angles are congruent. So, the final statement (statement 13) can be \( \triangle QWU \cong \triangle STV \) (SSS) and then \( \angle Q\cong\angle S \) (CPCTC). But since the problem says "Complete the proof that \( \angle Q\cong\angle S \)", the last statement should be \( \angle Q\cong\angle S \) with reason CPCTC, and the prior statement is the triangle congruence.

Wait, maybe a simpler way: After proving \( QW = ST \), \( QU = SV \), and \( UW = TV \), we can say \( \triangle QWU\cong\triangle STV \) (SSS), then \( \angle Q\cong\angle S \) (CPCTC). So, in the table:

Statement 13: \( \triangle QWU \cong \triangle STV \)
Reason: SSS (Side - Side - Side) Congruence Postulate

Statement 14: \( \angle Q\cong\angle S \)
Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

But since the table has up to 13 (with an empty 13th row), maybe the 13th row is \( \triangle QWU \cong \triangle STV \) (SSS) and then the angle congruence is implied. But to directly prove \( \angle Q\cong\angle S \), the statement is \( \angle Q\cong\angle S \) and the reason is CPCTC, with the triangle congruence as the prior step.

So, if we consider that the 13th statement is \( \angle Q\cong\angle S \), the reason is CPCTC (Corresponding Parts of Congruent Triangles are Congruent), and the triangles are congruent by SSS (from the side equalities we proved).

So, the final answer for the proof completion:

Statement 13: \( \angle Q\cong\angle S \)
Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent) (after proving \( \triangle QWU \cong \triangle STV \) by SSS)