ABCD is a trapezium in which AB parallel DC and its diagonals intersect each other at point O . show that AO/BO = CO/DO

A trapezoid (trapezium) is a quadrilateral in which one pair of opposite sides are parallel.

Answer: For a trapezium ABCD  in which AB is parallel to DC and its diagonals intersect each other at point O we see that AO/BO = CO/DO

Let's prove AO/BO = CO/DO

Explanation:

Let's draw the diagram of trapezium ABCD as shown below:

Here, the sides DC and AB are parallel sides.

Now, consider the two triangles AOB and COD

Since DC is parallel to AB,

∠ODC = ∠OBA (Alternate interior angles)

Similarly, ∠OCD = ∠OAB (Alternate interior angles)

Also, ∠COD = ∠AOB (Vertically opposite angles).

Now according to AAA (angle-angle-angle) similarity criteria, the two triangles AOB and COD are similar.

Therefore, AO/CO = BO/DO

Now, interchange the positions of CO and BO in the above equation.

Hence, AO/BO = CO/DO

Therefore, AO/BO = CO/DO