Question

Complete the following proof.

Answer 1

1-
`$\overline{QS} || \overline{RT}$`,
`$\angle R \cong \angle S$`; 2-
`$\angle QTS \cong \angle TQR$`; 3-
`$\angle SQT \cong \angle SQT$`; 4-
`$\triangle SQT \cong \triangle RTQ$`; 5-
`$\overline{QT} \cong \overline{TQ}$`.

Here's how the proof goes:

1. We are given that
`$\overline{QS} || \overline{RT}$` and
`$\angle R \cong \angle S$` (Statements 1 and 2).

2. From these two statements, we can conclude that
`$\angle QTS \cong \angle TQR$` using the alternate interior angles theorem (Statement 3).

3. Next, we use the reflexive property of congruence to get
`$\angle SQT \cong \angle SQT$`(Statement 4).

4. Now we have two pairs of congruent angles:
`$\angle QTS \cong \angle TQR$` and
`$\angle SQT \cong \angle SQT$`. Together with the given
`$\overline{QS} || \overline{RT}$`, this is enough to conclude that
`$\triangle SQT \cong \triangle RTQ$` by the angle-angle-side (AAS) congruence theorem (Statement 5).

5. Finally, from the corresponding parts of congruent triangles, we can conclude that
`$\overline{QT} \cong \overline{TQ}$` (Statement 6).

Therefore, we have proven that
`$\overline{QT} \cong \overline{TQ}$`, which is one of the defining properties of a parallelogram. Since we were also given that
`$\overline{QS} || \overline{RT}$`, we can conclude that the quadrilateral QTSR is a parallelogram.

In other words, the proof shows that if a quadrilateral has two opposite sides parallel and one pair of opposite angles congruent, then it must be a parallelogram.

This is a special case of the more general parallelogram definition, which states that a quadrilateral is a parallelogram if and only if both pairs of opposite sides are parallel.