question_answer

In the given figure, ABCD is a cyclic quadrilateral whose side AB is a diameter of the circle passing through A, B, C and D. If\[\angle ADC={{130}^{o}},\]find \[\angle BAC.\]

A) \[{{45}^{o}}\]

B)        \[{{58}^{o}}\]

C)        \[{{60}^{o}}\]

D)        \[{{40}^{o}}\]

Correct Answer: D

Solution :

Given, ABCD is a cyclic quadrilateral, \[\angle ADC={{130}^{o}}\] \[\therefore \]\[\angle ADC+\angle ABC={{180}^{o}}\] \[\Rightarrow \]\[\angle ABC={{50}^{o}}\] Since,  AB is a diameter of circle \[\therefore \]\[\angle ACB={{90}^{o}}\] Now, in \[\Delta ACB,\] \[\angle ABC+\angle ACB+\angle BAC={{180}^{o}}\] \[\Rightarrow \]\[{{50}^{o}}+{{90}^{o}}+\angle BAC={{180}^{o}}\Rightarrow \angle BAC={{40}^{o}}\]