# 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 A O B}={\angle C O D\,\,\, \text { (vertically opposite angles) }} \\ &{\angle B A O=\angle D C O\,\,\,( \text { alternate interior angles })} \\ &{\Rightarrow \Delta A O B \sim \Delta C O D\,\,\,(\text { AA criterion })} \\ \text{Hence,}\,\, { \frac{O A}{O C}=\frac{O B}{O D}}\end{align}

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