To prove the QRST is a parallelogram.
Step 1: Given [tex]\angle T \cong \angle R[/tex] and [tex]\overline{Q R} \| \overline{T S}[/tex]
Step 2: [tex]\angle R Q S \text { and } \angle T S Q[/tex] 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, [tex]\angle R Q S \cong \angle T S Q[/tex]
Step 4: Reflexive property of congruence:
Any geometric figure is congruence to itself.
Therefore, [tex]\overline{Q S} \cong \overline{Q S}[/tex]
Step 5: By the above steps
[tex]\angle T \cong \angle R[/tex] (Angle), [tex]\angle R Q S \cong \angle T S Q[/tex] (Angle) and [tex]\overline{Q S} \cong \overline{Q S}[/tex] (Side)
Hence [tex]\triangle Q T S \cong \triangle S R Q[/tex] (by ASA congruence theorem)
Step 6: Corresponding parts of congruence triangles are congruent.
[tex]\Rightarrow \ \overline{Q R} \cong \overline{T S}[/tex]
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.
