Question

given: $overline{vu}paralleloverline{qr}$, $overline{vt}paralleloverline{rs}$, $qt = us$ prove: $\triangle qrscong\triangle uvt$ 6. $qs = ut$ 7. given 8. $angle vtucongangle qsr$

Explanation:

Step1: Find corresponding angle 1

Since $\overline{VU}\parallel\overline{QR}$, corresponding angles are equal, so $\angle VUT=\angle QRT$.

Step2: Find corresponding angle 2

Since $\overline{VT}\parallel\overline{RS}$, corresponding angles are equal, so $\angle VTU=\angle QSR$.

Step3: Prove side - equality

Given $QT = US$, add $TS$ to both sides: $QT + TS=US+TS$, so $QS = UT$.

Step4: Prove triangle congruence

We have $\angle VUT=\angle QRT$, $QS = UT$, $\angle VTU=\angle QSR$. By ASA criterion, $\triangle QRS\cong\triangle UVT$.

Answer:

Since $\overline{VU}\parallel\overline{QR}$ and $\overline{VT}\parallel\overline{RS}$, we can use the properties of parallel - lines to find angle - equalities. Also, given $QT = US$, we can show the congruence of $\triangle QRS$ and $\triangle UVT$.

1. Because $\overline{VU}\parallel\overline{QR}$, $\angle VUT=\angle QRT$ (corresponding angles).
2. Because $\overline{VT}\parallel\overline{RS}$, $\angle VTU=\angle QSR$ (corresponding angles).
3. Given $QT = US$, then $QT+TS=US + TS$, so $QS = UT$.
4. In $\triangle QRS$ and $\triangle UVT$:

• $\angle VUT=\angle QRT$ (from step 1).
• $QS = UT$ (from step 3).
• $\angle VTU=\angle QSR$ (from step 2).
• By the Angle - Side - Angle (ASA) congruence criterion, $\triangle QRS\cong\triangle UVT$.