What is the measure of ∠dab? polygon abcd is inscribed in a circle. angle bcd is 122 degrees.

Check the picture attached.

BCD is an inscribed angle, so the measure of the arc BD (intercepted by angle BCD), is twice the measure of BCD.

So m arc (BAD)=2*122°=244°.

Let m(DAB)=α degrees, then m arc(BCD) is 2α degrees.

m arc (BAD) + m arc(BCD) =360°
 
[or  m major arc(BD) + m minor arc(BD) =360°]

so
244°+2α°=360°
2α°=360°-244°=116°
α°=116°/2=58°

m(BAD)=58°

Remark: In any quadrilateral inscribed in a circle, the opposite angles are supplementary, that is, the sum of their measures is 180°.

so given the angle with measure 122°, and asked for the measure of the opposite angle, by this theorem, we could have justified that the measure of the angle asked is 180°- 122°=58°

Also, if we are given that 2 opposite angles of a quadrilateral are supplementary, than we can circumscribe a circle to this quadrilateral.

What can we do about it