In the diagram, \(\overline{AD}\) is a diameter of a circle with Centre O. If ABD is a triangle in a semi-circle ∠OAB=34", find: (a) ∠OAB (b) ∠OCB

In the diagram, \(\overline{AD}\) is a diameter of a circle with Centre O. If ABD is a triangle in a semi-circle ∠OAB=34",

find: (a) ∠OAB (b) ∠OCB

Answer Details

(a) ∠OAB = 34°: Given in the problem.

(b) ∠OCB: We know that in a circle, opposite angles are equal. So, ∠OCB = ∠OAB = 34°.

Explanation: When a straight line cuts a circle at two points, it is called a chord of the circle. And when the chord is a diameter of the circle, it is called the diameter. The angle formed between the chord and the line drawn from the center of the circle to the midpoint of the chord is called the angle subtended by the chord at the center of the circle. And it is equal to half of the angle formed by the chord at the circumference of the circle.

So, in this case, ABD is a triangle inscribed in a semicircle. And ∠OAB is half of the central angle subtended by the chord AB. Hence, ∠OCB, which is opposite to ∠OAB is equal to ∠OAB.