# The diagonals of a quadrilateral ABCD intersect each other at the point O such thatShow that ABCD is a trapezium

The quadrilateral ABCD is shown below, BD and AC are the diagonals.

Construction: Draw a line OE parallel to AB

Given: In  ΔABD, OE is parallel to AB

To prove : ABCD is a trapezium
According to basic proportionality theorem, if in a triangle another line is drawn parallel to any side of triangle,
then the sides so obtain are proportional to each other.

Now, using basic proportionality theorem in ΔDOE and ΔABD, we obtain

...(i)

It is given that,

...(ii)

From (i) and (ii), we get

...(iii)
Now for ABCD to be a trapezium AB has to be parallel of CD

Now From the figure we can see that If eq(iii) exists then,

EO || DC (By the converse of basic proportionality theorem)

Now if,

AB || OE || DC

Then it is clear that

AB || CD

Thus the opposite sides are parallel and therefore it is a trapezium.

Hence,

ABCD is a trapezium

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