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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