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# In quadrilateral , sides and have equal length and sides and are parallel to each other. Point is the intersection of the diagonals

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In quadrilateral$$PQRS$$, sides $$\overline {PS}$$ and $$\overline {QR}$$ have equal length and sides $$\overline{PQ}$$ and $$\overline{RS}$$ are parallel to each other. Point $$X$$ is the intersection of the diagonals $$\overline {PR}$$ and $$\overline {QS}$$. Which of the following conclusions must be true?

A) $$\triangle PXQ\cong\triangle RXS$$

B) $$\triangle SPR\cong\triangle RQS$$

C) $$\triangle SPR\cong\triangle RQS$$

Type your answer as a list of letters separated by commas. For instance, if you believe that conditions A, B, and C are enough, then type "A,B,C" into the answer box. If you believe none of the options are correct, enter "none."

Sep 29, 2019

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Hmmmm.....it looks as though answer B and C are exactly the same....typo?

Sep 29, 2019
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We have this:

Angle QXR = angle SXP  (vertical angles)

Angle PSR = angle PQR

Angle PQS = angle RSQ  ( result of diagonal SQ cutting parallels PQ and SR)

Therefore.....Angle RQX  = angle PSX

PS = QR  (given)

So triangles QXR  is congruent to triangle SXP by AAS

Therefore..... XR = XP

Angle PXQ  = angle RXS   (vertical angles)

Angle XRS  = angle XPQ   ( PR is a transversal cutting parallels)

XR = XP

Therefore.....by ASA.....triangle PXQ is congruent to triangle RXS

Sep 30, 2019