Question

the figure shows quadrilateral pqrs. diagonals \\(\overline{pr}\\) and \\(\overline{qs}\\) intersect at point t.
choose all the conditions that prove that pqrs is a parallelogram.
\\(\square\\) a. \\(\overline{ps} \cong \overline{qr}\\)
\\(\square\\) b. \\(\overline{ps} \parallel \overline{qr}\\)
\\(\square\\) c. \\(\overline{ps} \parallel \overline{qr}\\) and \\(\overline{ps} \cong \overline{qr}\\)
\\(\square\\) d. \\(\overline{pq} \cong \overline{sr}\\) and \\(\overline{ps} \cong \overline{qr}\\)
\\(\square\\) e. \\(\overline{pq} \cong \overline{ps}\\) and \\(\overline{rq} \cong \overline{rs}\\)
\\(\square\\) f. \\(\overline{pt} \cong \overline{tr}\\) and \\(\overline{qt} \cong \overline{ts}\\)

Explanation:

Step1: Recall parallelogram theorems

A quadrilateral is a parallelogram if:

1. One pair of sides is both parallel and congruent.
2. Both pairs of opposite sides are congruent.
3. Diagonals bisect each other (segments of diagonals are congruent).

Step2: Evaluate each option

• A. Only one pair of sides congruent: Not sufficient.
• B. Only one pair of sides parallel: Not sufficient.
• C. One pair is parallel + congruent: Satisfies theorem 1.
• D. Both pairs of opposite sides congruent: Satisfies theorem 2.
• E. Adjacent sides congruent: Defines a kite, not parallelogram.
• F. Diagonals bisect each other: Satisfies theorem 3.

Answer:

C. $\overline{PS} \parallel \overline{QR}$ and $\overline{PS} \cong \overline{QR}$
D. $\overline{PQ} \cong \overline{SR}$ and $\overline{PS} \cong \overline{QR}$
F. $\overline{PT} \cong \overline{TR}$ and $\overline{QT} \cong \overline{TS}$