Let's solve this cyclic quadrilateral problem. We are given that quadrilateral ABCD is inscribed in a circle. The arc BC measures 90 degrees, and angle DBC equals 55 degrees. We need to find the measure of angle BCD.To solve this problem, we need to recall the inscribed angle theorem. This theorem states that an inscribed angle is equal to half the measure of its intercepted arc. This is a fundamental property of cyclic quadrilaterals and circles.Let's start by finding angle BAC. Since angle BAC is an inscribed angle that intercepts arc BC, we can apply the inscribed angle theorem. Angle BAC equals one half times the arc BC. Substituting the given value, angle BAC equals one half times 90 degrees, which gives us 45 degrees.Now let's analyze triangle ABC. We know that the sum of angles in any triangle equals 180 degrees. In triangle ABC, we have angle BAC plus angle ABC plus angle ACB equals 180 degrees. We just found that angle BAC is 45 degrees, so we can substitute this value into our equation.Notice that angle BCD can be decomposed into two parts: angle ACB and angle ACD. To find angle BCD, we need to determine the measures of both these angles. Let's work on finding each of them.Now let's use the given information that angle DBC equals 55 degrees. Looking at triangle DBC, we know that the sum of its angles must equal 180 degrees. So angle DBC plus angle BCD plus angle BDC equals 180 degrees. Substituting the known value, we get 55 degrees plus angle BCD plus angle BDC equals 180 degrees.Here's another crucial property of cyclic quadrilaterals: opposite angles are supplementary, meaning they add up to 180 degrees. In our quadrilateral ABCD, angle BAD plus angle BCD equals 180 degrees. Similarly, angle ABC plus angle ADC equals 180 degrees. This property will help us find the answer.To use the supplementary angle property, we need to find angle BAD. Notice that angle BAD can be split into angle BAC and angle CAD. We already know that angle BAC equals 45 degrees. Now we need to find angle CAD using the inscribed angle theorem.Let's try an alternative approach. Notice that angle ABC can be decomposed into angle ABD plus angle DBC. We know that angle DBC equals 55 degrees. Angle ABD is an inscribed angle that intercepts arc AD. This gives us another way to solve the problem.Let's use a more direct calculation method. Angle ACD equals one half times arc AD by the inscribed angle theorem. From triangle ABC, we can express angle ACB as 180 degrees minus 45 degrees minus angle ABC. This will help us find the components of angle BCD.Now let's calculate the solution. Angle DBC is an inscribed angle that intercepts arc DC. Since angle DBC equals 55 degrees, and it equals one half times arc DC, we can solve for arc DC. Multiplying both sides by 2, we get that arc DC equals 110 degrees.Now we can find arc BCD. Arc BCD equals arc BC plus arc CD. We know that arc BC equals 90 degrees, and arc CD equals 110 degrees. Adding these together, arc BCD equals 200 degrees. This is the total arc from B to D passing through C.Now we can find angle BAD using the inscribed angle theorem. Angle BAD is an inscribed angle that intercepts arc BCD. Therefore, angle BAD equals one half times arc BCD. Substituting our value, angle BAD equals one half times 200 degrees, which gives us 100 degrees.Finally, we can find angle BCD using the property that opposite angles in a cyclic quadrilateral are supplementary. Angle BAD plus angle BCD equals 180 degrees. Substituting angle BAD equals 100 degrees, we get 100 degrees plus angle BCD equals 180 degrees. Solving for angle BCD, we subtract 100 from 180, giving us angle BCD equals 80 degrees. This is our final answer.Let's summarize our solution. We used the inscribed angle theorem to find that angle BAD equals 100 degrees by calculating the arc BCD. Then we applied the cyclic quadrilateral property that opposite angles are supplementary. Finally, we calculated that angle BCD equals 80 degrees. This problem demonstrates the power of combining multiple geometric theorems to solve complex problems.