To find the measure of angle DBC, we can use the properties of circles and angles. Since BD is a diameter, we know that it divides the circle into two semicircles and any angle inscribed in a semicircle is a right angle (90 degrees).
The angle DBC is formed by the radius BC and the diameter BD. We also know that the measure of arc CB is 54 degrees. The measure of an inscribed angle is half the measure of the intercepted arc. Therefore:
Measure of angle DBC = 1/2 × Measure of arc CB
Substituting the known value:
Measure of angle DBC = 1/2 × 54 = 27 degrees
Thus, the measure of angle DBC is 27 degrees.