A tangent touches a circle with centre O at the point T, and P is a point on that tangent. TA is a chord of the circle drawn so that the acute angle between the chord TA and the tangent ray TP measures 40°. What is the measure of angle OTA?

A radius is perpendicular to the tangent at the point of contact, so angle OTP = 90°. The chord TA lies between the radius OT and the ray TP, so angle OTA = 90° − 40° = 50°. As a check, triangle OTA is isosceles with OT = OA, so angle OAT = 50° too and the central angle TOA = 80°, which is exactly twice the tangent–chord angle of 40°, as the alternate-segment theorem requires.