Circles: Special Angles and Arcs in Circles

Inscribed Angle: A Special Case

Let’s look at one special case of an inscribed angle.  This angle still has its vertex on the circle, but one side of the angle is a tangent line, rather than a chord.  The same property applies.  The measure of the angle is still one-half the measure of the intercepted arc.

Circle with point C and B on the circle, point A outside the circle, tangent line AB and chord CB, arc CB measures 164 degrees
Circle with chord and tangent

To find the measure of angle ABC, we notice that it is made up of a chord and a tangent and its vertex is on the circle.  The intercepted arc, arc C B, measures 164°.  The measure of the angle will be one-half the measure of the intercepted arc.

the measure of angle ABC equals one-half the measure of arc CB equals one-half of 164 equals 82 degrees