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.