Question

If ABCD is a cyclic quadrilateral with AB as the diameter of the circle and \(\angle ACD = 50^\circ,\) then \(\angle BAD\) = ___.

Answer 1

Since an angle in a semicircle is a right angle, \(\angle ACB=90^\circ.\)
But, \(\angle BCD=\angle ACD+\angle ACB=50^\circ+90^\circ=140^\circ.\)
Consider the cyclic quadrilateral ABCD. Since opposite angles are supplementary,
\(\angle BCD+\angle DAB=180^\circ.\)
\(\implies \angle BAD=180^\circ-140^\circ=40^\circ\)