question_answer

ABCD is a cyclic quadrilateral and AD is a diameter. If \[\angle DAC=55{}^\circ ,\]then the value of \[\angle ABC\]is

[SSC (CGL) Mains 2014]

A) \[55{}^\circ \]                          

B) \[35{}^\circ \]

C) \[145{}^\circ \]

D) \[125{}^\circ \]

Correct Answer: C

Solution :

In \[\Delta ACD,\]\[\angle DAC=55{}^\circ \]        [given]

\[\angle ACD=90{}^\circ =\]Angle in a semi-circle

\[\angle ADC=180{}^\circ -90{}^\circ -55{}^\circ \]

\[=180{}^\circ -145{}^\circ =35{}^\circ \]

Now, in a cyclic quadrilateral sum of opposite angles \[=180{}^\circ \]

\[\angle ABC+\angle ADC=180{}^\circ \]

\[\angle ABC=180{}^\circ -\angle ADC=180{}^\circ -35{}^\circ =145{}^\circ \]