Write a two column proof:
Given: ABCD is a parallelogram
BQ is conguent to DS, PA is congruent to RC
Prove PQRS is a parallelogram
Statement | Reason
----------------------------------------------
1. ABCD is a parallelogram | Given
2. BQ is congruent to DS | Given
3. PA is congruent to RC | Given
4. BC is parallel to AD | Opposite sides of a parallelogram are parallel (property of parallelograms)
5. BQ is parallel to DS | If two sides of a quadrilateral are congruent and parallel to each other, the quadrilateral is a parallelogram (property of parallelograms)
6. PQ is parallel to RS | Corresponding sides of congruent and parallel lines are congruent and parallel to each other (property of parallel lines)
7. AQ is parallel to CD | Opposite sides of a parallelogram are parallel (property of parallelograms)
8. PA is parallel to RC | Corresponding sides of congruent and parallel lines are congruent and parallel to each other (property of parallel lines)
9. PQRS Is a parallelogram | Opposite sides of a quadrilateral are parallel and congruent, it is a parallelogram (property of parallelograms)
To prove that PQRS is a parallelogram, we can use a two column proof. Each column will contain a statement and the reason for that statement. Here's the proof:
Statement | Reason
-------------------------
1. AB || CD | Given: ABCD is a parallelogram
2. BQ = DS | Given: BQ is congruent to DS
3. PA = RC | Given: PA is congruent to RC
4. ∠QAB = ∠SDC | Corresponding angles of parallel lines are congruent
5. ∠PAB = ∠RDC | Corresponding angles of parallel lines are congruent
6. ∠QAB + ∠PAB = 180° | Opposite angles of a parallelogram are supplementary
7. ∠SDC + ∠RDC = 180° | Opposite angles of a parallelogram are supplementary
8. ∠QAP = ∠SDR | Vertical angles are congruent
9. PQ || SR | Corresponding angles of parallel lines are congruent
10. PS || QR | Corresponding angles of parallel lines are congruent
11. PQRS is a parallelogram | Definition of a parallelogram
In this proof, we start by stating the given information: ABCD is a parallelogram, BQ is congruent to DS, and PA is congruent to RC. We then use the properties of parallelograms to show that the opposite angles (3 and 7) and corresponding angles (4, 5, 9, and 10) are congruent. By showing that the opposite angles are supplementary (6 and 7) and the corresponding angles are congruent, we conclude that PQRS is a parallelogram.