the parallelogram ABCD AC and BD are its diagonals intersect at p is the midpoint of deo and q is mid point of OB prove that a p c q is a parallelogram

the parallelogram ABCD AC and BD are its diagonals intersect at p is t

1.	the parallelogram ABCD AC and BD are its diagonals intersect at p is the midpoint of deo and q is mid point of OB prove that a p c q is a parallelogram
Answer» :-ABCD is a Parallelogram.Diagonals AC & BD bisects at O.p and q are the mid points of DO & BO RESPECTIVELY.◘ Required To Prove:-APCQ is a Parallelogram.◘ Proof:-By The Point of TrisectionAs, The diagonals of a Parallelogram Intrest each other, OB = ODAnd p & q are midpoints of OD & BO respectively. Now, Consider P & QBQ = PQ = DPNow,In ∆ADP & ∆BCQ→ AD = BC [Opposite Sides of a Parallelogram] → DP = BQ [Proved Above]→ ∠ADP = ∠CBQ [Alternative Interior ANGLES]∆ADP ≅ ∆BCQ [By SAS axiom]AP = CQ [By CPCT]Now,In ∆DCP & ∆BAQ→ DC = BA [Opposite Sides of Parallelogram]→ DP = BQ [Proved Above]→ ∠CDP = ∠ABQ [Alternative Interior Angles]∆CDP ≅ ∆ABQ [By SAS axiom]AQ = CP [By CPCT]As, The opposite sides are equal it MIGHT be a Rectangle or Parallelogram.Now, Consider OB & OD➝ OB = OD➝ BQ+OQ = OP+DP➝ BQ+OQ = OP+BQ [.:. BQ = DP]➝ OQ = OPIn Quadrilateral APCQOQ = OPOA = OCSince, the Diagonals of the given it is definitely a Parallelogram.A Quadrilateral in which two pairs of OPPOSITES sides are equal and diagonals bisect each other then it is a Parallelogram.Hence, APCQ is a Parallelogram.@MrSovereign ツHope This Helps!!