4 points A, B, C, D are taken on the circle with center O, so that AD is the diameter. Angle ACB = 12 °, angle BOC = 64 °. Find the CAD angle.

univerkov | November 19, 2020 | education

Points C and B lie on opposite sides of point A, since then the angle ACB can be less than 90º.
Moreover, AD passes through the middle of the CB side.
Angle CAD = CAO.
The VOS triangle is isosceles and its sides OS and OB are equal to the radius.
Consequently, angles C and B are equal in the BOC triangle.
Therefore, the angles ВСО = С = В = (180º – О) = (180º – 64º) / 2 = 58º.
Angle СОА = СОВ / 2 = 64º / 2 = 32º.
Consider the CAO triangle.
Angle АСО = ВСО + АСВ = 58º + 12º = 70º.
SAD angle = CAO = 180º- (COA + ASO) = 180º- (32º + 70º) = 180º- 102º = 78º.

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