Diagonals $AC$ and $BD$ of a trapezium $ABCD$ with $AB\ ∥\ DC$ intersect each other at the point $O$. Using similarity criterion for two triangles, show that $\frac{OA}{OC}\ =\ \frac{OB}{OD}$. "
Given:
Diagonals $AC$ and $BD$ of a trapezium $ABCD$ with $AB\ ∥\ DC$ intersect each other at the point $O$.
To do:
We have to show that $\frac{OA}{OC}\ =\ \frac{OB}{OD}$.
Solution:
In $\vartriangle AOB$ and $\vartriangle COD$,
$\angle AOB = \angle COD$ (Vertically opposite angles are equal)
$\angle OAB = \angle OCD$ ($AB\ ∥\ DC$, Alternate angles)
Therefore,
$\vartriangle AOB ∼ \vartriangle COD$
This implies,
$\frac{OA}{OC} = \frac{OB}{OD}$ (Corresponding sides are proportional)
Hence proved.
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