Inscribed Angles and Cyclic Quadrilaterals

An inscribed angle is an angle formed by two chords in a circle that have a common endpoint on the circle. The key principle to remember is that the measure of an inscribed angle is exactly half the measure of its intercepted arc. If the arc measures $x^\circ$, the inscribed angle measures $\frac{1}{2}x^\circ$. Conversely, if two inscribed angles intercept the same arc, they must be equal. This explains why a soccer player moving along an arc maintains the same 'view' of the goal—the angle remains constant because the intercepted arc (the goal width) hasn't changed.

A cyclic quadrilateral is a four-sided figure where all four vertices lie on the circumference of a circle. Because the four angles of the quadrilateral intercept arcs that combine to form a full $360^\circ$ circle, a unique property emerges: opposite angles are supplementary. This means they add up to $180^\circ$. For a quadrilateral $ABCD$ inscribed in a circle, $\angle A + \angle C = 180^\circ$ and $\angle B + \angle D = 180^\circ$. This property is essential for structural engineering when designing circular supports.

When lines intersect a circle, the angle they form depends on where they meet. 1. Inside the circle: The angle is half the sum of the intercepted arcs: $m\angle = \frac{1}{2}(arc_1 + arc_2)$. 2. Outside the circle: (formed by two secants, two tangents, or a secant and a tangent), the angle is half the difference of the intercepted arcs: $m\angle = \frac{1}{2}(arc_{far} - arc_{near})$. Think of it as the 'average' when inside and the 'gap' when outside.