# Ex.6.3 Q3 Triangles Solution - NCERT Maths Class 10

## Question

Diagonals \(AC\) and \(BD\) of a trapezium \(ABCD\) with \( AB\, ||\, DC\) intersect each other at the point \(O.\) Using a similarity criterion for two triangles, show that \(\begin{align}\frac{OA}{OC}=\frac{OB}{OD}\end{align}\).

**Diagram**

## Text Solution

**Reasoning:**

If two angles of one triangle are respectively equal to the two angles of another triangle, then the two triangles are similar.

This is referred to as the \(AA\) criterion**.**

**Steps:**

In \(\Delta \rm{A O B}, \Delta \rm{C O D}\)

\[\begin{align} & \angle AOB=\angle COD \\ & \text{(Vertically}\,\text{opposite}\,\,\text{angles)}\\ \\ & \angle BAO=\angle DCO \\ & (\text{Alternate interior angles }) \\\\ & \Rightarrow \Delta AOB\sim\Delta COD \\ & (\text{AA criterion}) \\ \end{align}\]

Hence \(\begin{align} \frac{O A}{O C}=\frac{O B}{O D}\end{align}\)