Question

Please help, I'm not sure if I'm right.

Answer 1

To prove the QRST is a parallelogram.

Step 1: Given $\angle T \cong \angle R$ and $\overline{Q R} \| \overline{T S}$

Step 2: $\angle R Q S \text { and } \angle T S Q$ are alternate interior angles.

Definition of alternate interior angles

Step 3: Alternate interior angle theorem:

If two parallel lines cut by a transversal then the alternate interior angles are congruence.

QR and TS are parallel lines cut by QS.

Therefore, $\angle R Q S \cong \angle T S Q$

Step 4: Reflexive property of congruence:

Any geometric figure is congruence to itself.

Therefore, $\overline{Q S} \cong \overline{Q S}$

Step 5: By the above steps

$\angle T \cong \angle R$ (Angle), $\angle R Q S \cong \angle T S Q$ (Angle) and $\overline{Q S} \cong \overline{Q S}$ (Side)

Hence $\triangle Q T S \cong \triangle S R Q$ (by ASA congruence theorem)

Step 6: Corresponding parts of congruence triangles are congruent.

$\Rightarrow \ \overline{Q R} \cong \overline{T S}$

Step 7: Property of parallelogram:

If one pair of opposite sides are both parallel and congruent, then the quadrilateral is a parallelogram.

Hence Quadrilateral QRST is a parallelogram.