Arcs and Inscribed Angles of Circles Inscribed angles have: Vertex on the circle Sides are chords (Chords AB and BC) Angle ABC is inscribed in the circle AC is the intercepted arc because it is created by Chords AB and BC Inscribed Angle Theorem: The measure of an inscribed angle is half the measure of its intercepted arc. Angle ABC = ½ AC Corollaries to the inscribed angle theorem: 1.) Two inscribed angles that intercept the same arc are congruent. 2.) An angle inscribed in a semi-circle is a right angle. 3.) The opposite angles of a quadrilateral inscribed in a are supplementary. circle
An angle whose vertex lies on a circle and whose sides intercept the circle (the sides contain chords of the circle) is called an inscribed angle. The measure of an inscribed angle is half the measure of the arc it intercepts. Figure 1: The inscribed angle measures half of the arc it intercepts If the vertex of an angle is on a circle, but one of the sides of the angle is contained in a line tangent to the circle, the angle is no longer an inscribed angle. The measure of such an angle, however, is equal to the measure of an inscribed angle. It is equal to one-half the measure of the arc it intercepts.
Figure 2: An angle whose sides are a chord and a tangent segment The angle ABC is equal to half the measure of arc AB (the minor arc defined by points A and B, of course). An angle whose vertex lies in the interior of a circle, but not at its center, has rays, or sides, that can be extended to form two secant lines. These secant lines intersect each other at the vertex of the angle. The measure of such an angle is half the sum of the measures of the arcs it intercepts. Figure3: An angle whose vertex is in the interior of a circle The measure of angle 1 is equal to half the sum of the measures of arcs AB and DE. When an angle's vertex lies outside of a circle, and its sides don't intersect with the circle, we don't necessarily know anything about the angle. The angle's sides, however, can intersect with the circle in three different ways. Its sides can be contained in two secant lines, one secant line and one tangent line, or two tangent lines. In any case, the measure of the angle is one-half the difference between the measures of the arcs it intercepts. Each case is pictured below.
Figure 4: An angle whose vertex lies outside of a circle In part (A) of the figure above, the measure of angle 1 is equal to one-half the difference between the measures of arcs JK and LM. In part (B), the measure of angle 2 is equal to one-half the difference between the measures of arcs QR and SR. In part (C), the measure of angle 3 is equal to one-half the difference between the measures of arcs BH and BJH. In this case, J is a point labeled just to make it easier to understand that when an angle's sides are parts of lines tangent to a circle, the arcs they intercept are the major and minor arc defined by the points of tangency. Here, arc BJH is the major arc. Central angles are probably the angles most often associated with a circle, but by no means are they the only ones. Angles may be inscribed in the circumference of the circle or formed by intersecting chords and other lines. Inscribed angle: In a circle, this is an angle formed by two chords with the vertex on the circle.
Intercepted arc: Corresponding to an angle, this is the portion of the circle that lies in the interior of the angle together with the endpoints of the arc. In Figure 1, ABC is an inscribed angle and is its intercepted arc. Figure 1 An inscribed angle and its intercepted arc. Figure 2 shows examples of angles that are not inscribed angles. Figure 2 Angles that are not inscribed angles. Figure 3 A circle with two diameters and a (nondiameter) chord. Notice that m 3 is exactly half of m, and m 4 is half of m 3 and 4 are inscribed angles, and and are their intercepted arcs, which leads to the following theorem.
Theorem 70: The measure of an inscribed angle in a circle equals half the measure of its intercepted arc. The following two theorems directly follow from Theorem 70. Theorem 71: If two inscribed angles of a circle intercept the same arc or arcs of equal measure, then the inscribed angles have equal measure. Theorem 72: If an inscribed angle intercepts a semicircle, then its measure is 90. Example 1: Find m C in Figure 4. Figure 4 Finding the measure of an inscribed angle. Example 2: Find m A and m B in Figure 5. Figure 5 Two inscribed angles with the same measure. Example 3: In Figure 6, QS is a diameter. Find m R. m R = 90 (Theorem 72). Figure 6 An inscribed angle which intercepts a semicircle. Example 4: In Figure 7 of circle O, m 60 and m 1 = 25.
Figure 7 A circle with inscribed angles, central angles, and associated arcs. Find each of the following. a. m CAD b. m c. m BOC d. m e. m ACB f. m ABC