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."
+36916 Hmmmm.....it looks as though answer B and C are exactly the same....typo?
+128474 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
