__Day 7 Lines Intersecting
Inside or Outside a Circle__

If
two lines intersect a circle, there are three places where the lines can
intersect.

You
know how to find the angle and arc measures when lines intersect *on* the circle. Now we’ll examine theorems that help us to
find measures when the lines intersect *inside*
or *outside* the circle.

1^{ST} Theorem:

If two chords intersect in the *interior* of a circle, then

the measure of each angle is one
half the *sum* of the

measures of the arcs intercepted by
the angle and its vertical angle.

Proof:

An
angle formed by a secant segment and a tangent to a circle is called a __secant-tangent
angle__. The next theorem involves
secant-tangent angles.

__CASE I__

If a tangent
and a secant intersect in the *exterior*
of a circle,

then the
measure of the angle formed is one half the *difference
*

of the
measures of the intercepted arcs.

Proof:

A __tangent-tangent angle__ is
the angle formed by two tangents to a circle. The following theorem involves the measurement
of the tangent-tangent angle.

If two
tangents intersect in the *exterior* of
a circle,

then the measure of the angle
formed is one half the *difference *

of the measures of the intercepted
arcs.

Proof:

The angle formed when two secants
intersect is a __secant angle__. This
last theorem looks at the measurement of the secant angle.

If two
secants intersect in the *exterior* of
a circle,

then the measure of the angle
formed is one half the *difference *

of the measures of the intercepted
arcs.

Proof: