question_answer

The diagonals AC and BD of a parallelogram ABCD intersect each other at the point O. If \[\angle DAC={{32}^{o}}\]and \[\angle AOB={{70}^{o}},\]then,\[\angle DBC\] is equal to ____.            

A) \[{{38}^{o}}\]

B)        \[{{86}^{o}}\]

C)        \[{{24}^{o}}\]

D)        \[{{32}^{o}}\]

Correct Answer: A

Solution :

Given, \[\angle DAC={{32}^{o}}\]As \[DA||BC\]and AC is transversal. \[\therefore \]\[\angle ACB=\angle DAC={{32}^{o}}\] (alternate angles) Also, \[\angle AOB+\angle BOC={{180}^{o}}\](linear pair) \[\Rightarrow \]\[{{70}^{o}}+\angle BOC={{180}^{o}}\] \[\Rightarrow \]\[\angle BOC={{110}^{o}}\] In \[\Delta BOC,\angle BOC+\angle OBC+\angle OCB={{180}^{o}}\] (angle sum property) \[\Rightarrow \]\[{{110}^{o}}+\angle OBC+{{32}^{o}}={{180}^{o}}\] \[\Rightarrow \]\[\angle OBC={{180}^{o}}-({{110}^{o}}+{{32}^{o}})={{38}^{o}}\] \[\Rightarrow \]\[\angle DBC={{38}^{o}}\]