Idiot\'s Guides Basic Math and Pre-Algebra

206 Part 3: The Shape of the World

When an angle is formed by two tangents, the circle is broken into two arcs. When two secants

make the angle, there are four arcs, but only the two that fall between the secants are used in

the measurement. When a secant and a tangent team up to make an angle, you get three arcs,

ACCD,, and AD, as shown below. The two you want are the two between the tangent and the

secant, AC and AD. The measure of ’F is

1

2

AC AD.

WORLDLY WISDOM

All the rules about the measurement of angles in and around circles can be boiled

down to four rules, according to where the vertex of the angle is. If the vertex is at the

center, the angle equals its intercepted arc. If the vertex is on the circle, the angle is

half the intercepted arc. If the vertex is inside the circle, the angle is half the sum of the

arcs, and if the vertex is outside the circle, the angle is half the difference of the arcs.

One last type of angle that you should know about is the angle that’s formed by a line and a

segment, or specifically a tangent and a radius or diameter, or a tangent and a chord.

A tangent just touches the circle at one point. When you draw a chord from that point of

tangency T to some other point on the circle, like C, you divide the circle into two arcs, minor

arc TC and major arc TDC, and you create two angles, ’ATC and ’BTC. The measure of each

angle is half its intercepted arc. m ATC TD 51 C

2

and m BTC TD 51 C

2

. The two arcs add up to
the whole circle, or 360r, so half of each one will add to 180r, which sounds just right because the
angles are a linear pair and linear pairs are supplementary.

A

F

D

C