State true or false:If the diagonals of a quadrilateral divide each other proportionally, then it is a trapezium.

A
True
B
False

Solution:

The line parallel to one side of the triangle that intersects the other two sides in two distinct points then the other two sides of the triangle are divided in the same ratio.
The required figure is,

As the diagonals of quadrilaterals divide each other proportionally. So,

\frac{AO}{OC} = \frac{BO}{OD}

Cross multiply the above equation.

\frac{AO}{BO} = \frac{OC}{OD}

………(1)
Thus,

∠ AOB = ∠ COD

(Vertically opposite angles) ………(2)
So, by SAS similarity criterion,

ΔAOB \sim ΔCOD

We know that the corresponding angles of similar triangles are equal.
As

ΔAOB \sim ΔCOD

so,

∠ ABD = ∠ BDC

And,

∠ CAD = ∠ ACB

.
Since, the pairs of alternate angles are equal, thus

AB | | CD

.
So, the given quadrilateral is trapezium.
Therefore, it is true that if the diagonals of a quadrilateral divide each other proportionally, then it is a trapezium.
Hence, option 1 is correct.