Inscribed Angle
GeometryAn 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 corner (vertex) sitting ON the circle, with its two sides as chords of the circle. An inscribed angle is always half the size of the central angle that opens to the same arc.
Example
If an arc is $80^\circ$, the central angle for that arc is $80^\circ$, but any inscribed angle opening to the same arc is only $40^\circ$. All inscribed angles opening to the same arc are equal.
Key Insight
All inscribed angles that "look at" the same arc are equal - no matter where on the circle the vertex is placed. This is a remarkable fact: the view angle stays the same as you move around the circle.
Definition
An inscribed angle is formed by two chords meeting at a point on the circle. Inscribed Angle Theorem: the inscribed angle equals half the intercepted arc (= half the central angle for the same arc). Corollary: all inscribed angles intercepting the same arc are equal. Thales' theorem is the special case: inscribed angle in a semicircle $=$ half of $180$ $= 90^\circ$.
Example
Arc $AB$ measures $110^\circ$. Central angle for $AB = 110^\circ$. Any inscribed angle with vertex on the major arc intercepting arc $AB = 110/2 = 55^\circ$. Inscribed angle with vertex on the minor arc intercepting the major arc $= (360-110)/2 = 125^\circ$.
Key Insight
The inscribed angle theorem is used to prove that opposite angles of a cyclic quadrilateral are supplementary: the two inscribed angles each intercept the arcs that together make the full circle ($360^\circ$), so they sum to $360/2 = 180^\circ$.
Definition
The inscribed angle theorem states that the inscribed angle $\alpha$ subtending arc of measure $2\alpha$ (the central angle). Proof: for central angle $O$ and inscribed angle $A$, draw the radius through $A$; using isosceles triangles (all radii equal) and exterior angle theorem, $\alpha = \theta/2$. Generalizes to: any angle formed by two chords intersecting inside the circle $= (1/2)(\text{sum of intercepted arcs})$.
Example
Two chords $PQ$ and $RS$ intersect at $T$ inside a circle. Angle $PTR = (1/2)(\text{arc } PR + \text{arc } QS)$. If arc $PR = 80^\circ$ and arc $QS = 40^\circ$, then angle $PTR = 60^\circ$. The supplementary angle $PTS = (1/2)(\text{arc } PS + \text{arc } QR) = 120^\circ$.
Key Insight
The inscribed angle theorem is the key to understanding why cyclic quadrilaterals have supplementary opposite angles, why Ptolemy's theorem holds, and why the angles in a triangle inscribed in a circle relate to the triangle's circumradius via the law of sines $a/\sin A = 2R$. The single theorem about inscribed angles underlies a large part of circle geometry.