A tangent to a circle
is a line in the plane of the circle that intersects the circle in
exactly one point.

If you spin an object in a
circular orbit and release it, it will travel on a path that
is tangent to the circular orbit.

Theorem:

If a line is tangent to a circle, it is
perpendicular to the radius drawn to the point of tangency.

Theorem:

Tangent segments to a circle from the
same external point are congruent.
(You may think of this as the "Hat" Theorem because
the diagram looks like a circle wearing a pointed hat.)

This theorem can be proven
using congruent triangles and the previous theorem.
The triangles shown below are congruent by the Hypotenuse
Leg Postulate for Right Triangles. The radii (legs)
are congruent and the hypotenuse is shared by both
triangles. By using Corresponding Parts of Congruent
Triangles are Congruent, this theorem is proven true.

Common Tangents:

Common tangents are lines or segments
that are tangent to more than one circle at the same time.