Chapter 5.2, Problem 11WE

To determine

To prove: that $TQRS$ is a parallelogram with the help of given information.

Answer to Problem 11WE

  $TQRS$ is a parallelogram.

Explanation of Solution

Given information:

  $TQRS$ is a quadrilateral.

  $\begin{array}{l}TS\cong QR\\ TQ\cong SR\end{array}$

Calculation:

Considering the above information,

To Prove: $TQRS$ is a parallelogram

  

Proof: The two column proof is given below:

Statements	Reasons
In $△TSQ$ and $△RQS$ $\begin{array}{l}TS\cong QR\\ TQ\cong SR\end{array}$	Given and it follows from the statement of the theorem.
$QS\cong SQ$	Common side to both triangles
$△TSQ\cong △RQS$	S.S.S Test
$\begin{array}{l}\angle TSQ\cong \angle RQS\\ \angle TQS\cong \angle RSQ\end{array}$	Corresponding parts of congruent triangles
But $\angle TSQ\cong \angle RQS$ Then $TS\parallel QR$	Alternate interior angles
$\begin{array}{l}\angle TQS\cong \angle RSQ\\ \text{Then }TQ\parallel SR\end{array}$	It is also alternate interior angles
$TQRS$ is a parallelogram	If opposite side in quadrilateral are parallel then that quadrilateral is parallelogram.

Hence, the above statements prove that $TQRS$ is a parallelogram.