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°