Central Angle and Inscribed Angle Relationship

What is the relationship between central angle and inscribed angle in a circle?

Given that the diameter of circle O is AC, and angle MBC is 62 degrees, what is the measure of angle AOB?

Explanation:

The relationship between central angle and inscribed angle in a circle is that the measure of an inscribed angle is half the measure of the central angle subtended by the same arc. In this case, we are given that angle MBC is a central angle with a measure of 62 degrees.

When dealing with a circle, a central angle is an angle formed by two radii drawn from the center of the circle to any two points on its circumference. In contrast, an inscribed angle is an angle formed by two chords or tangents intersecting on the circumference of the circle.

For inscribed angles, the measure of the angle is equal to half the measure of the central angle that subtends the same arc. This relationship holds true for any circle, regardless of the size or radius.

In the given scenario, we are asked to find the measure of angle AOB, which is the inscribed angle in circle O. Since angle MBC is the central angle subtended by the same arc, we can apply the relationship mentioned above to calculate the measure of angle AOB.

By multiplying the measure of angle MBC (62 degrees) by 2, we can find the measure of angle AOB. Therefore, the measure of angle AOB is 124 degrees.

Knowing the relationship between central angles and inscribed angles can help solve various problems related to angles within circles and arcs. It provides a fundamental understanding of the geometric properties of circles and their associated angles.