The segment BD is the diameter of the circle with the center O, the chord AC bisects the radius of the OB

The segment BD is the diameter of the circle with the center O, the chord AC bisects the radius of the OB and is perpendicular to it. Find the angles ABCD and the degree measures of the arcs AB, BC, CD, AD.

Let’s construct an additional radius OA.

The ACO triangle is rectangular since AC is perpendicular to BD. By condition, point K is the middle of the OB radius, then the leg OK is half of the hypotenuse OA, and then the angle OAK = 30, and the angle KOA = 60.

The AOD angle is adjacent to the KOA angle, then AOD = (180 – 60) = 120.

Triangle AOD is isosceles, then the angle OAD = ODA = (180 – 120) / 2 = 30.

Angle ВAK = 90 – ОAC – OAD = 90 – 30 – 30 = 30. Angle AВK = 90 – 90 = 60.

Similarly, in a triangle ВСD, angle ВDC = 30, angle DВС = 60.

Then the angle ABC = 120, angle ADC = 60, angle BAD = BCD = 90.

Arc AB = 60, BC = 60, CD = 120, AD = 120.

Answer: The angles of the quadrilateral are 60, 60, 120, 120.

The arcs are equal: AB = 60, BC = 60, CD = 120, AD = 120.