Question

in triangle PQR median PM is produced to X such that PM = MX Prove that PQXR is a parallelogram

Answer 1

Refer to the above attachment.
Hope it helps you....

Answer 2

Proved below.  

Step-by-step explanation:

Given: ​

ΔPQR with median PM to side QR. PM is extended to X such that PM = MX.

Construction: Join XR.  

Let PQXR be a parallelogram.    

Therefore consider ​ΔPQM and ​ΔXRM  

PM = MX {Given}

∠PMQ = ​∠RMX       {Vertically Opposite Angles are equal}

QM = MR                 {Given that PM is a median hence it will bisect QR}

Therefore ΔPQM  ≅​ ​ΔXRM  {by SAS criteria}  

Hence, PQ = RX by CPCT                 (1)  

Also, ∠PQR = ∠XRM by CPCT         (2)

From Eqn. (2) we can infer that PQ is parallel to XR, since alternate interior angles (∠PQR and ∠XRM) are equal.

Hence PQ | | XR                                  (3)

From (1) and (2) we can say that PQXR is a parallelogram since in quadrilateral PQXR one pair of opposite sides are equal and parallel.