Inscribed Angle
An inscribed angle has its vertex on a circle with sides that are chords, measuring exactly half the central angle that intercepts the same arc.
Formula
\text{inscribed angle} = (1/2) \times \text{intercepted arc}
Definition
An inscribed angle has its vertex on a circle with its sides as chords; the Inscribed Angle Theorem states it always equals half the intercepted arc (half the central angle for the same arc), so all inscribed angles intercepting the same arc are equal, no matter where the vertex sits on the circle. Thales' theorem is the special case where the arc is a semicircle, giving an inscribed angle of $90^\circ$. More generally, an angle formed by two chords intersecting inside a circle equals half the sum of the two intercepted arcs, and one formed by two chords meeting on the circle proves that a cyclic quadrilateral's opposite angles are supplementary, since their intercepted arcs together make the full $360^\circ$ circle.
Example
If an arc is $80^\circ$, its central angle is $80^\circ$ but any inscribed angle opening to it is only $40^\circ$, and this holds no matter where on the circle the vertex sits. If arc $AB$ measures $110^\circ$: an inscribed angle on the major arc intercepting it is $55^\circ$, while one on the minor arc intercepting the major arc is $(360-110)/2=125^\circ$. Two chords $PQ$ and $RS$ intersecting at $T$ inside a circle give angle $PTR=(1/2)(\text{arc }PR+\text{arc }QS)$: if these arcs are $80^\circ$ and $40^\circ$, angle $PTR=60^\circ$.
Key Insight
All inscribed angles that "look at" the same arc are equal, a remarkable fact since the view angle stays the same no matter where you stand on the circle. The proof draws a radius through the inscribed angle's vertex and uses isosceles triangles (all radii equal) with the exterior angle theorem to show the inscribed angle is half the central angle, and this single theorem underlies a large part of circle geometry, from why cyclic quadrilaterals have supplementary opposite angles, to Ptolemy's theorem, to the law of sines $a/\sin A=2R$ relating a triangle's angles to its circumradius.