Question

given: ( overline{st} parallel overline{qr} ); ( overline{qr} cong overline{st} )
prove: ( overline{qt} cong overline{rs} )

Explanation:

Step1: Reason for $\overline{ST}\parallel\overline{QR}$

Given.

Step2: Reason for $\overline{QS}\cong\overline{QS}$

Reflexive property of congruence (a segment is congruent to itself).

Step3: Reason for $\angle QST\cong\angle RQS$

Alternate - interior angles theorem (if two parallel lines are cut by a transversal, then alternate - interior angles are congruent. Here, $\overline{ST}\parallel\overline{QR}$ and $\overline{QS}$ is the transversal).

Step4: Reason for $\triangle QST\cong\triangle SQR$

SAS (Side - Angle - Side) congruence criterion. We have $\overline{ST}\cong\overline{QR}$ (given), $\angle QST\cong\angle RQS$ (from step 3), and $\overline{QS}\cong\overline{QS}$ (from step 2).

Step5: Reason for $\overline{RS}\cong\overline{QT}$

CPCTC (Corresponding Parts of Congruent Triangles are Congruent). Since $\triangle QST\cong\triangle SQR$, their corresponding sides $\overline{RS}$ and $\overline{QT}$ are congruent.

Answer:

The reasons are: Given (for $\overline{ST}\parallel\overline{QR}$), Reflexive property (for $\overline{QS}\cong\overline{QS}$), Alternate - interior angles theorem (for $\angle QST\cong\angle RQS$), SAS (for $\triangle QST\cong\triangle SQR$), CPCTC (for $\overline{RS}\cong\overline{QT}$)