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.

1ST 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: