Question

The diagonal of a rectangle ABCD intersect each other at O. If $\angle AOB={30}^o$, find $\angle COD$ and $\angle OCD$

• A. $\angle COD={30}^o,\angle OCD={75}^o$
• B. $\angle COD={40}^o,\angle OCD={75}^o$
• C. $\angle COD={30}^o,\angle OCD={65}^o$
• D. none of the above

Answer 1

The correct option is A $\angle COD={30}^o,\angle OCD={75}^o$

From figure,
$\angle COD=\angle AOB$ (Vertically opposite angles)
$⟹\angle COD={30}^{\circ }$
In $△OCD$,
$\angle OCD+\angle ODC+\angle COD={180}^{\circ }$ ---------(1)

$AC=BD$ $[$ Diagonals are equal in rectangle $]$

$⟹\frac{AC}{2}=\frac{BD}{2}$

$⟹OC=OD$ $[$ Diagonals of rectangle intersect each other at mid-points $]$

$⟹\angle OCD=\angle ODC$ $[$ Angles opposite to equal sides are equal $]$
$\angle OCD+\angle OCD+\angle COD={180}^{\circ }$ $[$ Using (1) $]$
$⟹2\angle OCD+{30}^{\circ }={180}^{\circ }$
$⟹\angle OCD={75}^{\circ }$