Let *A *and *B* be points on a
circle. If the
line that goes through *A *and *B *goes through the center
of the circle, we say
that *A *and *B *are **diametically opposed**.

If *A *and *B *are diametrically opposed, then the
line through *A *and *B *divides
the circle into two
**half circles**.

Let *A *and *B *be two points on a
circle
centered at *O*. The **arc** between *A *and *B *is the set of points on the
circle which are in the
angle between *OA *and *OB*.

If *A *and *B *are diametrically
opposed, we will have to specify which
half circle we mean.

Let *A *and *B *be two points on a
circle. A
**partition** of the arc between *A *and *B *is a
set of points

on the arc between *A *and *B *ordered so that *P*_{i }is in the
angle between *OP*_{i-1 }and *OP _{i}*

The **inside approximation of
the arc length** from *A *to *B *corresponding to the partition *A* = *P*_{0} , *P*_{1}, . . . , *P _{n}* =

Let *A *and *B *be two points on a
circle. The **outside
approximation of the arc length** from *A *to *B *corresponding to the partition *A* = *P*_{0} , *P*_{1}, . . . , *P _{n}* =

Where *T _{i}* is the point where the tangent at

One partition is
called a **refinement** of another
partition if every
point in one
partition is also a
point in the other one.

Let *A *and *B *be two points on a
circle. The length of the
arc between *A *and *B *or the **arc length**
between *A *and *B *is the least upper bound of all of the inside
approximations.^{1}

The **unit circle** is the
circle of
radius 1 centered at the origin.

In the
angle between *AB *and *AC*, the ray from *A *thorough *B *and the ray from *A *through *C *are called the **arms** of the
angle.

The angle illustrated
above is called __/ __*BAC*. The middle letter indicates the **vertex** of the
angle, and the other two
letters are points on the two
arms of the
angle.

For any
angle, the ratio of the
arc length of the arc of
a circle
centered at the
vertex of the
angle bewtween the
arms of the
angle to the
radius of the
circle is called the
number of **radians** in the angle, or the
**radian measure of the
angle**.^{2} The notation for the number of radians in an angle __/ __*BAC *will be *m*__/ __*BAC*.

To get the number of **degrees** in an angle, multiply the number of radians by 180^{o}/*pi*^{3}

A **straight angle** is one whose
two rays are opposite
rays of the same straight
line.

If the arms of an
angle are
perpendicular, the
angle is called a **right angle**.

The number of radians in a straight angle is denoted by pi. Pi is a Greek letter that looks like

A straight
angle has 180 **degrees**.

An angle which is
smaller than a right angle is called an
**acute** angle.

An angle which is
between a right angle and a
straight angle in size is called an
**obtuse** angle.

If two angles
add up to a straight angle, they are
called **supplementary**.

If two angles
add up to a right angle they are called
**complementary**.

If two
lines cross in a
plane as in the figure below,
the angles which are opposite each other called
**vertical angles**.

A **rotation** is accomplished by fixing
one point of the
plane and moving all the
points through a fixed angle about the fixed point.

In the figure, *A *is moved to *A*', *B *is moved to *B*', and *C *is
moved to *C*'. In order for the movement to be a rotation we must have

and __/ __*AOA*', __/ __*BOB*', and __/ __*COC*' all have the same size.

If you start at a
point on a
circle where a horizontal line through the
center meets the
circle in the negative *x *direction and proceed in the positive *y *direction, you are going **clockwise.**

The opposite direction along a
circle from clockwise is
called **counterclockwise**.

Analytic Foundations of Geometry