Circle Theorems
Some interesting things about angles and circles.
Inscribed Angle
First off, a definition:
Inscribed Angle: an angle made from points sitting on the circle's circumference.
A and C are "end points"
B is the "apex point"
Play with it here:
When you move point "B", what happens to the angle?
Inscribed Angle Theorems
An inscribed angle a° is half of the central angle 2a°
(Called the Angle at the Center Theorem)
And (keeping the end points fixed) ...
... the angle a° is always the same,
no matter where it is on the same arc between end points:
Angle a° is the same.
(Called the Angles Subtended by Same Arc Theorem)
Example: What is the size of Angle POQ? (O is circle's center)
Angle POQ = 2 × Angle PRQ = 2 × 62° = 124°
Example: What is the size of Angle CBX?
Angle ADB = 32° also equals Angle ACB.
And Angle ACB also equals Angle XCB.
So in triangle BXC we know Angle BXC = 85°, and Angle XCB = 32°
Now use angles of a triangle add to 180° :
Angle in a Semicircle (Thales' Theorem)
An angle inscribed across a circle's diameter is always a right angle:
(The end points are either end of a circle's diameter,
the apex point can be anywhere on the circumference.)
Why? Because: The inscribed angle 90° is half of the central angle 180° (Using "Angle at the Center Theorem" above) 
Another Good Reason Why It Works
We could also rotate the shape around 180° to make a rectangle!
It is a rectangle, because all sides are parallel, and both diagonals are equal.
And so its internal angles are all right angles (90°).
So there we go! No matter where that angle is
on the circumference, it is always 90°
Example: What is the size of Angle BAC?
The Angle in the Semicircle Theorem tells us that Angle ACB = 90°
Now use angles of a triangle add to 180° to find Angle BAC:
Finding a Circle's Center
We can use this idea to find a circle's center:
 draw a right angle from anywhere on the circle's circumference, then draw the diameter where the two legs hit the circle
 do that again but for a different diameter
Where the diameters cross is the center!
Drawing a Circle
When we know two opposite points on a circle we can draw that circle.
Put some pins or nails on those points and use a builder's square like this:
Cyclic Quadrilateral
A "Cyclic" Quadrilateral has every vertex on a circle's circumference: 

A Cyclic Quadrilateral's opposite angles add to 180°:

Example: What is the size of Angle WXY?
Opposite angles of a cyclic quadrilateral add to 180°
Tangent AngleA tangent line just touches a circle at one point. It always forms a right angle with the circle's radius. 