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"
Inscribed Angle Theorems
An inscibed angle a° is half of the central angle 2a°
(Called the Angle at the Center Theorem)
And (keeping the endpoints fixed) ...
... the angle a° is always the same, no matter where it is on the circumference:
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° equals Angle ACB. And Angle ACB 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 CBX + Angle BXC + Angle XCB = 180°
Angle CBX + 85° + 32° = 180°
Angle CBX = 63° 
Angle in a Semicircle
An angle inscribed in a semicircle 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 inscibed angle 90° is half of the central angle 180° (Using "Angle at the Center Theorem" above) 
Another Good Reason Why It WorksWe 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: Angle BAC + 55° + 90° = 180°
Angle BAC = 35° 
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° Angle WZY + Angle WXY = 180°
69° + Angle WXY = 180°
Angle WXY = 111° 
Tangent AngleA tangent is a line that just touches a circle at one point. It always forms a right angle with the circle's radius. 