Given that ∠CAD = ∠DBC = 55     (Since angles in the same segment)

∠DAB = ∠CAD + ∠BAC

         = 55 + 45

         = 100

similarly ∠DAB + ∠BCD = 180   (Opposite angle of a cyclic quadilateral)

 => ∠BCD = 180 - ∠DAB

                = 180 - 100

=> ∠BCD = 80

Problem Statement:
In cyclic quadrilateral ABCD, AC and BD are the diagonals. Angle DBC is given as 55 degrees and angle BAC is given as 45 degrees. We need to find the measure of angle BCD.

Solution:
Given:
Angle DBC = 55 degrees
Angle BAC = 45 degrees

Properties of a Cyclic Quadrilateral:
A cyclic quadrilateral is a quadrilateral in which all four vertices lie on a circle. It has several important properties:

1. Opposite angles are supplementary: The sum of opposite angles in a cyclic quadrilateral is always 180 degrees.
2. Angle at the center: The angle formed by two lines originating from the center of the circle and intersecting the circumference is double the angle formed by those lines on any part of the circumference.
3. Angles in the same segment: Angles in the same segment of a circle are equal.

Identifying the Required Angle:
To find angle BCD, we need to identify a relationship involving this angle and the given angles.

Using Angle at the Center Property:
Let O be the center of the circle passing through the points A, B, C, and D. Since ABCD is a cyclic quadrilateral, angle BAC and angle BCD are subtended by the same arc AC. Therefore, we can use the angle at the center property to relate these angles.

Angle BCD = 2 * Angle BAC
Angle BCD = 2 * 45 degrees
Angle BCD = 90 degrees

Answer:
Therefore, the measure of angle BCD is 90 degrees.