To solve for the measure of angle EBC in circle O, where AC and BE are diameters, we need to follow a few steps based on the properties of circles and the relationships between arcs and angles.
1. **Understanding Diameters and Arcs**: Since AC and BE are diameters of circle O, this means they divide the circle into four quadrants. The arc DC is a part of the circle that is measured between points D and C along the circumference.
2. **Measure of Arc DC**: It is given that the measure of arc DC is 50 degrees. This arc lies between points D and C on the circle.
3. **Finding Measure of Arc EC**: Because AC is a diameter, the entire circle is 360 degrees. The opposite arc, which is arc EC, complements arc DC. Hence, we can find arc EC by subtracting the measure of arc DC from 180 degrees (since D and C are endpoints of the diameter line AC):
Arc EC = 180 degrees – Arc DC
Arc EC = 180 degrees – 50 degrees = 130 degrees
4. **Using Inscribed Angle Theorem**: The Inscribed Angle Theorem states that the measure of an inscribed angle is half the measure of the arc it intercepts. Here, angle EBC intercepts arc EC. Therefore, we can determine the measure of angle EBC as:
Measure of Angle EBC = 1/2 * Arc EC
Measure of Angle EBC = 1/2 * 130 degrees = 65 degrees
In conclusion, the measure of angle EBC is 65 degrees.