Question

$angle rqv cong angle stu$, $angle qvr cong angle tsu$, and $overline{uv} cong overline{rs}$. complete the proof that $\triangle stu cong \triangle vqr$.
1 $angle rqv cong angle stu$ given
2 $angle qvr cong angle tsu$ given
3 $overline{uv} cong overline{rs}$ given
4 $rv = uv + ru$ additive property of length
5 $su = rs + ru$
6 $rv = rs + ru$ substitution
7 $su = rv$
8 $\triangle stu cong \triangle vqr$

Explanation:

Step1: Identify reason for Step5

The statement $SU = RS + RU$ uses the same logic as Step4, which is the additive property of segment length, since segment $SU$ is composed of segments $RS$ and $RU$.

Step2: Identify reason for Step7

We know $RV = RS + RU$ (Step6) and $SU = RS + RU$ (Step5). By substituting the equal expressions, we can conclude $SU = RV$, so the reason is substitution.

Step3: Identify reason for Step8

We have two pairs of congruent angles: $\angle RQV \cong \angle STU$, $\angle QVR \cong \angle TSU$, and a pair of congruent included sides $\overline{SU} \cong \overline{RV}$. This matches the Angle-Side-Angle (ASA) congruence criterion for triangles.

Answer:

Statement	Reason
2. $\angle QVR \cong \angle TSU$	Given
3. $\overline{UV} \cong \overline{RS}$	Given
4. $RV = UV + RU$	Additive Property of Length
5. $SU = RS + RU$	Additive Property of Length
6. $RV = RS + RU$	Substitution
7. $SU = RV$	Substitution
8. $\Delta STU \cong \Delta VQR$	ASA Congruence Criterion