Why is $x$ not equal to $75$ degrees?

Question

Someone told me,

Draw $B C$. Let $\angle D B C=y$. Then $30+2 y=180 \Rightarrow y=75$. Now, notice that ABCD is a cyclic quadrilateral. As a result, $x+y=180 \Rightarrow x=105$ but

I cannot see their logic.

Answer 1

The following may be easier: Since $\text{m}\angle BDC = 30$, the arc $BC$ must span twice that, or $60$ degrees. Then, by symmetry, we have arc $CD$ spanning $\frac{360-60}{2} = 150$ degrees, so arc $BCD$ spans $210$ degrees. Therefore, $\text{m}\angle BAD = \frac{210}{2} = 105$ degrees.

Answer 2

Draw diameter from D , it meets the circle at E. we have:

$\overset{\frown}{EB}=\frac 12 [\overset{\frown}{EB}=60^o]=30^o$

$\Rightarrow \overset{\frown}{BCA}=30+180=210^o$

$\Rightarrow x=\frac 12\overset{\frown}{BCA}=\frac {210}2=105^o$