suppose a quadrilateral abcd,ad=bc and ac=bd.prove that it is a trapezium

given; ABCD is a quadrilateral in which AD=BC and AC = BD
TPT: ABCD is a trapezium.
proof: 
in the triangles ABD and BAC;
AB is common.
AD = BC
BD = AC
therefore by SSS congruency triangles are congruent.
therefore by CPCT; $\angle A=\angle B$............(1)
similarly in the triangles BDC and ACD;
CD is common.
BD = AC
AD = BC
therefore by SSS congruency triangles are congruent.
and by CPCT $\angle C=\angle D$ ...........(2)
since sum of interior angles of a quadrilateral is 360 deg.
$$\begin{gathered} \angle A+\angle B+\angle C+\angle D=360 \\ \angle A+\angle A+\angle D+\angle D=360 \\ 2*(\angle A+\angle D)=360 \\ \angle A+\angle D=180 \end{gathered}$$
since the sum of adjacent interior angles are 180 deg.
therefore AB || DC
thus ABCD is a trapezium.

hope this helps you.