question_answer

The diagonals of a rectangle PQRS intersect at O. If \[\angle ROQ={{60}^{o}},\]then find \[\angle OSP.\]

A) \[{{70}^{o}}\]

B)        \[{{50}^{o}}\]

C)        \[{{60}^{o}}\]

D)        \[{{80}^{o}}\]

Correct Answer: C

Solution :

\[\angle ROQ=\angle SOP={{60}^{o}}\]                        ?(i)      [Vertically opposite angles] \[\therefore \]\[PR=SQ\Rightarrow PO=SO\] (Diagonals of a rectangle are equal and bisect each other) \[\Rightarrow \]\[\angle OPS=\angle OSP\] [\[\because \]In a triangle, angles opposite to equal sides are equal] In \[\Delta POS,\]by angle sum property \[\angle OSP+\angle OPS+\angle SOP={{180}^{o}}\] \[\Rightarrow \]\[2\angle OSP={{180}^{o}}-{{60}^{o}}\][Using (i) & (ii)] \[\Rightarrow \]\[\angle OSP={{60}^{o}}\]