Find the angles of a quadrilateral ABCD inscribed in a circle if the angle CBD = 48 °, angle ACD = 34 °, angle BDC = 64 °.

Two angles are known in the BCD triangle, then the angle BCD = (180 – CBD – BDC) = (180 – 48 – 64) = 68.

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

Then the angle BAD = 180 – BCD = (180 – 68) = 112.

Angle АСВ = (ВСD – АСD) = (68 – 34) = 34. The inscribed angle АСВ rests on the arc AB as well as the inscribed angle АDВ, then the angle АDВ = АСВ = 340. Then the angle АDC = (АDВ + ВDC) = (34 + 64) = 98.

Then the opposite angle ABC = (180 – ADC) = (180 – 98) = 82.

Answer: The angles of the quadrilateral are 82, 86, 112, 98.