Ex.6.3 Q3 Triangles Solution - NCERT Maths Class 10

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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 \(\begin{align}\frac{OA}{OC}=\frac{OB}{OD}\end{align}\).


Text Solution


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.


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

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