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 ABTo 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
It is given that,
From (i) and (ii), we get
Now for ABCD to be a trapezium AB has to be parallel of CD
EO || DC (By the converse of basic proportionality theorem)Now if,
⇒ AB || OE || DCThen it is clear that
⇒ AB || CDThus the opposite sides are parallel and therefore it is a trapezium.
ABCD is a trapezium
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