Circles: Special Angles and Arcs in Circles

Inscribed Angles

An inscribed angle is an angle with its vertex on the circle.  The measure of an inscribed angle equals one-half the measure of its intercepted arc.  The measure of the intercepted arc is twice the measure of in inscribed angle.


Take a look at circle X below.

Circle X with points S, T and U on the circle, angle STU is drawn
Circle X with inscribed angle STU

If the measure of arc SU equals 100 degrees, then the measure of angle STU equals one-half of 100 equals 50 degrees

If the measure of angle STU equals 65 degrees, then the measure of arc SU equals 2 times 65 equals 130 degress

Example:
Circle W has the following properties:

arc HJ equals the quantity 22 x plus 50, degrees; the measure of angle HIJ equals the quantity 5 x plus 55, degrees

 

Circle W with points H, I and J on the circle, angle HIJ is drawn
Circle W with inscribed angle HIJ

 

Find the value of x.

Solution:

the measure of angle HIJ equals one-half the measure of arc HJ; 5 x plus 55 equals one-half the quantity 22 x plus 50

5x + 55 = 11x + 25
5x + 30 = 11x
30 = 6x
5 = x