Question

complete the proof that $\angle qrv \cong \angle s$.

(image of triangles with markings and a table with statements and reasons:

1. $\overline{qr} \cong \overline{st}$ (given)
2. $\overline{rs} \cong \overline{uv}$ (given)
3. $\overline{qv} \cong \overline{tu}$ (given)
4. $rv = uv + ru$ (additive property of length)
5. $su = rs + ru$ (additive property of length)
6. $rv = rs + ru$ (substitution)
7. $su = rv$ ( blank )
8. $\triangle qrv \cong \triangle tsu$ ( blank )
9. $\angle qrv \cong \angle s$ ( blank ))

Explanation:

Step1: Analyze Step 7

We have \( SU = RS + RU \) (from step 5) and \( RV = RS + RU \) (from step 6). By the Transitive Property of Equality (if \( a = b \) and \( b = c \), then \( a = c \)), since \( SU = RS + RU \) and \( RV = RS + RU \), we can conclude \( SU = RV \). So the reason for step 7 (\( SU \cong RV \)) is the Transitive Property of Equality (or Substitution, but more precisely Transitive since we're equating two quantities both equal to \( RS + RU \)).

Step2: Analyze Step 8

We have \( \overline{QR} \cong \overline{ST} \) (step 1), \( \overline{QV} \cong \overline{TU} \) (step 3), and now \( \overline{SU} \cong \overline{RV} \) (step 7). By the Side - Side - Side (SSS) Congruence Criterion, if three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. So \( \triangle QRV \cong \triangle TSU \) by SSS.

Step3: Analyze Step 9

Since \( \triangle QRV \cong \triangle TSU \) (step 8), corresponding parts of congruent triangles are congruent (CPCTC). So \( \angle QRV \cong \angle S \) because they are corresponding angles of congruent triangles.

Answer:

• Step 7 Reason: Transitive Property of Equality (or Substitution, as \( SU = RS + RU \) and \( RV = RS + RU \) implies \( SU = RV \))
• Step 8 Reason: SSS (Side - Side - Side) Congruence Criterion
• Step 9 Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)