The quadrilateral ABCD is inscribed in a circle. The ABC angle is 100 degrees and the CAD 69 angle is.

The quadrilateral ABCD is inscribed in a circle. The ABC angle is 100 degrees and the CAD 69 angle is. Find the angle ABD.

1. By the condition of the problem given: the inscribed angles ABC are 100 °, and CAD 69 °.

2. It is known:

a) any pair of angles based on the same chord and with vertices on opposite sides of it add up to 180 °;

b) the inscribed angles based on the same arc are equal,

c) the sum of the angles of the triangle is 180 °.

Hence the angle ABC + angle ADC = 180 °, whence the angle ADC = 180 ° – 100 ° = 80 °.

In triangle ACD, angle ACD = 180 ° – (angle CAD + angle ADC) = 180 ° – (69 ° + 80 °) = 180 ° – 149 ° = 31 °.

From the drawing, we see that the angles ACD and ABD are based on one arc AD and are therefore equal to 31 °.

Answer: Angle ABD 31 °.