Question

given: $overline{vu}paralleloverline{qr}$, $overline{vt}paralleloverline{rs}$, $qt = us$. prove: $\triangle qrscong\triangle uvt$. 7. given $angle vcongangle u$. 7. given. 8. $angle vtucongangle qsr$. 8. if two parallel lines are cut by a trans - versal, then their corresponding angles are congruent. 9. click here to insert. 9. angle - angle - side

Explanation:

Step1: Identify given parallel lines

$\overline{VU}\parallel\overline{QR}$ and $\overline{VT}\parallel\overline{RS}$, $QT = US$.

Step2: Use parallel - line property for angles

Since $\overline{VT}\parallel\overline{RS}$, by the corresponding - angles postulate, $\angle VTU\cong\angle QSR$ (when two parallel lines are cut by a transversal).

Step3: Consider given angle

We are given that $\angle V\cong\angle U$.

Step4: Find the third - angle relationship

We know that $QT = US$. Then $QT+TS=US + TS$, so $QS = UT$.
In $\triangle QRS$ and $\triangle UVT$, we have $\angle VTU\cong\angle QSR$, $\angle V\cong\angle U$, and $QS = UT$.
By the Angle - Angle - Side (AAS) congruence criterion, $\triangle QRS\cong\triangle UVT$.

Answer:

We have proven that $\triangle QRS\cong\triangle UVT$ by the Angle - Angle - Side (AAS) congruence criterion.