Angles formed by chords of a circle are known as inscribed angles:
Inscribed angle in a circle
An angle which is formed in the interior of a circle when two chords share an endpoint.
Multiple inscribed angles which share segments can form an inscribed polygon:
Inscribed polygon in a circle
A polygon which has all of its vertices on a circle.
When a polygon is inscribed in a circle, we can equivalently say that the circle is circumscribed on the polygon.
A circle can also be inscribed into a polygon:
Inscribed circle in a polygon
The largest possible circle that can be drawn in the interior of a polygon. If it is a regular polygon, then each side of the polygon is tangent to the circle.
Again, we can refer to this equivalently by saying that the polygon is circumscribed on the circle.
The following theorems relate to angles and polygons inscribed in circles:
Inscribed angle theorem
If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.
Congruent inscribed angle theorem
If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
Inscribed semicircle theorem
An angle inscribed in a semicircle is a right angle.
Inscribed right triangle theorem
If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.
Opposite inscribed angle theorem
The opposite angles of an inscribed quadrilateral are supplementary.
The opposite inscribed angle theorem says that two opposite angles in a quadrilateral inscribed in a circle add up to 180 \degree.
In this case, that results in the following equations:
m\angle BCE + m\angle EDB = 180
m\angle DBC + m\angle CED = 180
