The quadrilateral ABCD is inscribed in a circle. The ABD angle is 85 degrees, the CAD angle is 19

The quadrilateral ABCD is inscribed in a circle. The ABD angle is 85 degrees, the CAD angle is 19 degrees. Find the angle ABC.

Consider a quadrilateral ABCD inscribed in a circle.

By the condition of the problem, it is known that the angle ABD = 85 °, and the angle CAD = 19 °.

Note that the angles CAD and CBD rest on the same arc CD.

Therefore, these angles are equal to each other: CAD = CBD = 19 °.

Also note that the angle ABC is equal to the sum of the angles ABD and CBD:

ABC = ABD + CBD.

Hence it follows that:

ABC = ABD + CBD = 85 ° + 19 ° = 104 °.

Answer: The degree measure of the angle ABC = 104 °.