In order to solve problems which involve secants, tangents, and angles formed by them, it is necessary to

A typical problem involving the angles formed by secants and tangents in a

circle gives us information about the measures of the

angle exterior to the

circle and/or about the measures of the intercepted arcs of the circle. Two examples of this type of problem are presented below.

- In circle O shown below, two secants from point P intercept arcs CB = x – 10 and AD = 2x. What is the measure of arc AD if angle P is 25°?

We know that the measure of an external

angle P when formed by two secants is equal to one half the difference of the measures of the intercepted arcs.

25 = (1/2)(x + 10)

50 = x + 10

x = 40º

Since we were given that

arc AD = 2x

AD = 2(40º) = 80°

- In circle O shown below, angle P is x°, arc CB is 55°, and arc CD is 4x – 9. What is the measure of arc CD?

We know that

angle P must equal ½ the difference of the measures of arcs CD and CB.

2x = 4x – 64

2x = 64

x = 32

Since we were given that

arc CD = 4x – 9

CD = 4(32) – 9

CD = 119°