A tangent BA and a secant BD are drawn from point B to the circle. Find the value of the angle ABD

A tangent BA and a secant BD are drawn from point B to the circle. Find the value of the angle ABD if the arcs cut by them on the circle are equal to 209 ° and 97 °.

Let us define the degree measure of the arc CD. ◡СD = 360 – 97 – 209 = 54.

Consider a triangle COD, in which the central angle COD is equal to the magnitude of the degree measure of the arc CD.

Angle СОD = 54. Sides ОC and ОD are the radii of the circle, and therefore, triangle СОD is isosceles, ОC = ОD. Then the angle ОD = ОDС = (180 – 54) / 2 = 63.

Determine the value of the angle ОCB. Angle OCB = 180 – 63 = 117.

The angle OAB = 90, as the perpendicular from the center of the circle to the tangent.

Consider a quadrilateral BCOA, the sum of the angles of which is 360. Then the angle ABC = 360 – OAB – AOC – OCB = 360 – 90 – 97 – 117 = 56.

Angle AED = ABC = 56.

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