Question : ABCD is a cyclic quadrilateral and AD is a diameter. If $\angle$ DAC = 55$^\circ$ then value of $\angle$ABC is:
Option 1: 55$^\circ$
Option 2: 35$^\circ$
Option 3: 145$^\circ$
Option 4: 125$^\circ$
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Correct Answer: 145$^\circ$
Solution : Given, a cyclic quadrilateral ABCD and diameter AD And, $\angle$ DAC = 55$^\circ$ In $\triangle$ ACD, $\angle$ DAC = 55$^\circ$ $\angle$ ACD = 90$^\circ$ (angle in a semicircle) $\angle$ ADC = 180$^\circ$ – 55$^\circ$ – 90$^\circ$ = 35$^\circ$ (angle sum property) We know that opposite angles in a cyclic quadrilateral are supplementary. So, $\angle$ ABC+ $\angle$ ADC = 180$^\circ$ Or, $\angle$ ABC = 180$^\circ$ – 35$^\circ$ = 145$^\circ$ Hence, the correct answer is 145$^\circ$.
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