Answer:(a) AOD = 90 degrees
(b) BCD = 46 degrees
(c) DBO = 44 degrees
Step-by-step explanation:In the given diagram, let's find the measures of the angles AOD, BCD, and DBO.
(a) AOD: Since O is the center of the circle, the radius OA and OD are equal. In a circle, the angle formed by two radii is always 90 degrees. Therefore, the measure of angle AOD is 90 degrees.
(b) BCD: In a circle, the measure of an angle formed by two intersecting chords is half the sum of the intercepted arcs. In this case, the intercepted arc BCD is 92 degrees (40 degrees + 52 degrees). So, the measure of angle BCD is half of 92 degrees, which is 46 degrees.
(c) DBO: The angle DBO is an exterior angle to triangle OBD. By the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. In this case, the interior angles are OBD and BOD.
Since the sum of the angles in a triangle is always 180 degrees, we can find the measure of BOD by subtracting the measures of angle OBD and angle BDO from 180 degrees. Angle OBD is 90 degrees (since it's a right angle) and angle BDO is 46 degrees (as found in part (b)).
So, the measure of angle BOD is 180 degrees - 90 degrees - 46 degrees, which equals 44 degrees.
