Some interesting things about angles and circles.
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 inscribed 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° 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
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.)
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:
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°
A tangent is a line that just touches a circle at one point.
It always forms a right angle with the circle's radius.