Quadrilateral ABCD is inscribed in a circle. The angle ABC is 56 degrees, the CAD angle is 42 degrees. Find the angle ABC.

First way.

Since the quadrilateral is inscribed in a circle, the sum of its opposite angles is 180.

Then the angle ADС = 180 – ABC = 180 – 56 = 124.

In the AСD triangle, the AСD angle = 180 – 42 – 124 = 14.

Angle AВD = AСD = 14, since they are based on one arc of the AD.

Second way.

The inscribed angle СAD = 420 and rests on the arc of the СD, then the degree measure of the arc of the СD = 2 * 42 = 84.

The inscribed angle ABC = 560 and rests on the ADС arc, then the degree measure of the ADС arc = 2 * 56 = 112.

Then the degree measure of the arc AD = ᵕADС – ᵕСD = 112 – 84 = 28.

The inscribed angle of the AВD is based on the AD arc and is equal to half of its degree measure. AВD angle = 28/2 = 14.

Answer: The value of the AВD angle is 14.