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

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## 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\tilde{\ }\Delta COD \\ & (\text{ AA criterion }) \\ \end{align}

Hence $${ \frac{O A}{O C}=\frac{O B}{O D}}$$

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