Angles of a Circles

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Central angles and inscribed angles

An angle whose vertex is the center of the circle is called a central angle. The degree of the circle (the slice of pie) cut by a central angle is equal to the measure of the angle. If a central angle is 25º, then it cuts a 25º arc in the circle.

An inscribed angle is an angle formed by two chords in a circle that originate from a single point. An inscribed angle will always cut out an arc in the circle that is twice the size of the degree of the inscribed angle. If an inscribed angle has a degree of 40º, it will cut an arc of 80º in the circle.

If an inscribed angle and a central angle cut out the same arc in a circle, the central angle will be twice as large as the inscribed angle.

Properties of Angles in Circles

a. The angle subtended by an arc at the center is double the angle subtended by the same arc a. at the circumference of the circle.

b. Angles in the same segment are equal

c. The angle in a semi-circle is a right angle

d. The angle in a major segment is an acute angle and the angle in the minor segment is an obtuse angle.