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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
 #1
avatar+19879 
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Hmmmm.....it looks as though answer B and C are exactly the same....typo?

 Sep 29, 2019
 #2
avatar+106539 
+2

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

 

 

cool cool cool

 Sep 30, 2019

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