Question

given ( overline{qv} cong overline{st} ), ( overline{rs} cong overline{uv} ), and ( overline{qr} cong overline{tu} ). complete the proof that ( angle qrv cong angle sut ).

statement	reason
2 ( overline{rs} cong overline{uv} )	given
3 ( overline{qr} cong overline{tu} )	given
4 ( rv = uv + ru )	additive property of length
5 ( su = rs + ru )	additive property of length
6 ( rv = rs + ru )	substitution
7 ( su = rv )	transitive property of equality
8

Explanation:

Step1: Identify Triangle Congruence

We need to prove \( \angle QRV \cong \angle SUT \). First, check the sides: \( QV \cong ST \) (Given), \( RS \cong UV \) (Given), and we need to show \( QR \cong SU \) to use SAS (Side - Angle - Side) congruence.

Step2: Prove \( QR \cong SU \)

• From step 4: \( RV = UV + RU \) (Additive Property of Length)
• From step 2: \( RS = UV \) (Given), substitute \( RS \) for \( UV \) in step 4: \( RV=RS + RU \) (Substitution, step 6)
• From step 5: \( SU = RS+RU \) (Additive Property of Length)
• By Transitive Property of Equality (step 7: \( SU = RV \)), and from step 1: \( QV = ST \), step 2: \( RS = UV \), and now \( QR=QV - RV \), \( SU = ST - SU \)? Wait, no, let's re - evaluate. Wait, actually, we can use the SSS (Side - Side - Side) or SAS. Wait, let's list the sides:
• \( QV\cong ST \) (Given, step 1)
• \( RS\cong UV \) (Given, step 2)
• We need to show \( QR\cong SU \). From step 5: \( SU = RS + RU \), step 6: \( RV=RS + RU \), so \( SU = RV \) (step 7). Also, if we consider the triangles \( \triangle QRV \) and \( \triangle SUT \):
• \( QV\cong ST \) (Given)
• \( RV\cong SU \) (from \( SU = RV \))
• \( QR\cong TU \) (Given, step 3: \( QR\cong TU \)) Wait, no, step 3 is \( QR\cong TU \). Wait, maybe I made a mistake. Let's start over.
• The goal is \( \angle QRV\cong\angle SUT \). Let's look at the triangles \( \triangle QRV \) and \( \triangle SUT \).
• We know \( QV\cong ST \) (step 1), \( RS\cong UV \) (step 2), and we need to find the third side. From step 5: \( SU = RS + RU \), step 4: \( RV=UV + RU \), and since \( RS = UV \) (step 2), then \( SU = RV \) (by substituting \( RS \) with \( UV \) in step 5 and using step 4). So now we have:
• \( QV\cong ST \) (side)
• \( RV\cong SU \) (side, from \( SU = RV \))
• \( QR\cong TU \) (step 3, given) Wait, no, that's not right. Wait, maybe it's SAS. Let's check the angles. Wait, no, the correct approach is:
• In \( \triangle QRV \) and \( \triangle SUT \):
• \( QV\cong ST \) (Given)
• \( \angle QVR \) and \( \angle STU \)? No, wait, let's use the sides we have. Since \( QV\cong ST \), \( RS\cong UV \), and \( SU\cong RV \) (from \( SU = RS + RU \) and \( RV=RS + RU \)), then by SSS (Side - Side - Side) congruence criterion, \( \triangle QRV\cong\triangle SUT \).
• If \( \triangle QRV\cong\triangle SUT \), then \( \angle QRV\cong\angle SUT \) (Corresponding Parts of Congruent Triangles are Congruent - CPCTC)

Step8: Conclusion

After establishing that \( \triangle QRV\cong\triangle SUT \) (by SSS, since \( QV\cong ST \), \( RV\cong SU \), \( QR\cong TU \)), we can conclude that \( \angle QRV\cong\angle SUT \) by CPCTC (Corresponding Parts of Congruent Triangles are Congruent). So the last step (step 8) should be \( \angle QRV\cong\angle SUT \) with the reason "CPCTC (Corresponding Parts of Congruent Triangles are Congruent)".

Answer:

The last statement (step 8) is \( \boldsymbol{\angle QRV\cong\angle SUT} \) with the reason "CPCTC (Corresponding Parts of Congruent Triangles are Congruent)".