Idiot\'s Guides Basic Math and Pre-Algebra

Chapter 15: Circles 205

The figure shows two secants that cross at P. One cuts the circle at B and then again at A. The
other crosses the circle at C and at D. You want the measurement of ’P. To create some angles
you do know how to measure, draw BD. That will make 'BDP and exterior angle ’ABD.
m’ABD = m’P + m’BDC, so m’P = m’ABD – m’BDC. Both ’ABD and ’BDC are
inscribed angles, so each is equal to half its intercepted arc. That means that the measure of ’P
is half of AD minus half of BC, or half of the difference between AD and BC. The measure of an
angle formed by two secants is half the difference of the two arcs it intercepts.

Angles formed by two secants, a tangent and a secant, or two tangents will intercept two arcs.
The arc nearest to the vertex of the angle is the smaller of the two. The two secants, as you can
see in the figure, cut off a small arc BC the first time they cross the circle and a larger one AD
the second time they cross the circle. The measure of the angle is one-half the difference of the
two arcs it intercepts.

Two tangents are drawn to the circle from point P, as shown in the figure. Each tangent touches
the circle at one point, A or B. The circle is broken into two arcs: the minor arc AB and the major
arc ACB. The measure of ’P will be half the difference between the major arc and the minor arc,

m P^5 mACB^2 mAB
1
2

.

B

C

A

D

P

A

O

P

B

C