Question
The figure shows quadrilateral P Q R S. Diagonals segment P R and segment Q S intersect at point T.
A picture of quadrilateral P Q R S.
Question
Choose ,begin emphasis,all,end emphasis, the conditions that prove that P Q R S is a parallelogram.
Answer options with 6 options
A.
segment P S is congruent to segment Q R
B.
segment P S is parallel to segment Q R
C.
segment P S is parallel to segment Q R and segment P S is congruent to segment Q R
D.
segment P Q is congruent to segment S R and segment P S is congruent to segment Q R
E.
segment P Q is congruent to segment P S and segment R Q is congruent to segment R S
F.
segment P T is congruent to segment T R and segment Q T is congruent to segment T S
A picture of quadrilateral P Q R S.
Question
Choose ,begin emphasis,all,end emphasis, the conditions that prove that P Q R S is a parallelogram.
Answer options with 6 options
A.
segment P S is congruent to segment Q R
B.
segment P S is parallel to segment Q R
C.
segment P S is parallel to segment Q R and segment P S is congruent to segment Q R
D.
segment P Q is congruent to segment S R and segment P S is congruent to segment Q R
E.
segment P Q is congruent to segment P S and segment R Q is congruent to segment R S
F.
segment P T is congruent to segment T R and segment Q T is congruent to segment T S
Answers
GPT-4o mini
To prove that quadrilateral PQRS is a parallelogram, the following conditions are sufficient:
1. **Option A:** Segment PS is congruent to segment QR (If one pair of opposite sides is congruent, then the quadrilateral is a parallelogram.)
2. **Option B:** Segment PS is parallel to segment QR (If one pair of opposite sides is parallel, then the quadrilateral is a parallelogram.)
3. **Option C:** Segment PS is parallel to segment QR and segment PS is congruent to segment QR (If a pair of opposite sides is both congruent and parallel, then the quadrilateral is a parallelogram.)
4. **Option D:** Segment PQ is congruent to segment SR and segment PS is congruent to segment QR (If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram.)
5. **Option E:** Segment PQ is congruent to segment PS and segment QR is congruent to segment RS (This does not demonstrate that P Q R S is a parallelogram, as it does not focus on opposite sides being equal; instead, it compares adjacent sides.)
6. **Option F:** Segment PT is congruent to segment TR and segment QT is congruent to segment TS (This indicates that the diagonals are bisected but does not by itself establish that P Q R S is a parallelogram unless it also states that a pair of opposite sides is either parallel or congruent.)
Based on the above explanations, the options that will prove that PQRS is a parallelogram are **A, B, C, and D**.
1. **Option A:** Segment PS is congruent to segment QR (If one pair of opposite sides is congruent, then the quadrilateral is a parallelogram.)
2. **Option B:** Segment PS is parallel to segment QR (If one pair of opposite sides is parallel, then the quadrilateral is a parallelogram.)
3. **Option C:** Segment PS is parallel to segment QR and segment PS is congruent to segment QR (If a pair of opposite sides is both congruent and parallel, then the quadrilateral is a parallelogram.)
4. **Option D:** Segment PQ is congruent to segment SR and segment PS is congruent to segment QR (If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram.)
5. **Option E:** Segment PQ is congruent to segment PS and segment QR is congruent to segment RS (This does not demonstrate that P Q R S is a parallelogram, as it does not focus on opposite sides being equal; instead, it compares adjacent sides.)
6. **Option F:** Segment PT is congruent to segment TR and segment QT is congruent to segment TS (This indicates that the diagonals are bisected but does not by itself establish that P Q R S is a parallelogram unless it also states that a pair of opposite sides is either parallel or congruent.)
Based on the above explanations, the options that will prove that PQRS is a parallelogram are **A, B, C, and D**.