Question

1.Prove that a quadrilateral is a parallelogram if and only if the diagonals bisect each other.help please

Answer 1

First off, this condition is not 'If and only if'-there are multiple other ways to prove a quadrilateral to be a parallelogram. Anyway, back to the question:
Let the quadrilateral be ABCD.
Let the diagonals, AC and BD, meet at O.
Assume that the diagonals bisect each other,. Then, AO=OC and BO=OD.
In triangle AOB and DOC,
AO=OC
BO=OD
angle AOB=angle DOC.
thus by SAS congruency, AOB-COD (- denotes "is congruent to")
by Corresponding Parts of Congruent Triangles, ang. OBA=ang. ODC.
Thus AB|| CD ( with BD as transversal, interior opp. angles are equal, thus the two lines are parallel)
Similarly we can see that AD||BC.
Since pairs of opposite sides are parallel, the quad. ABCD is a parallelogram