AB is tangent to the circle k(O) at B, and AD is a secant, which goes through O. Point O is between A and D∈k(O). Find m∠BAD and m∠ADB, if measure of arc BD is 110°20'.

Answer 1

this would be so much easier for you to follow if i could supply you with a drawing. sorry i cannot.

since arc BD is 110°20', that means the central angle BOD is also 110°20'. angles AOD and BOD are a linear pair. so measure of angle AOB is 69 degrees 40 minutes.

any tangent line is perpendicular to the radius touching it, including line AB, so segment AB is perpendicular to segment BO, so angle ABO is 90 degrees.

the interior angles of any triangle sum to 180 degrees, so measure angle BAO + measure angle AOB + measure angle OBA = 180. that means measure angle BAO + 69 degrees 40 minutes + 90 = 180, so the measure of angle BAO (same as angle BAD) is 20 degrees 20 minutes.

segment AD goes thru the center of the circle, so that means the arc between the points where tangent line AD intersects the circle is half the circle, or 180 degrees. you already know arc BD is 110 degrees 20 minutes, so the remaining piece, between point B and the point on the circle closest to point A, is 180 degrees minus 110 degrees 20 minutes = 69 degrees 40 minutes. angle ADB is the same as the inscribed angle that intersects the 69 degrees 40 minutes arc, so it is half that size...34 degrees, 50 minutes.