Understanding Inscribed Angles and Arcs

In geometry, an inscribed angle is an angle formed by two chords in a circle that have a common endpoint on the circle. The arc that lies in the interior of the inscribed angle is called the intercepted arc. Understanding how these angles relate to the arcs they intercept is a fundamental part of circle geometry.

Key Theorems of Inscribed Angles

Here are the most important rules to remember when working with inscribed angles:

1. The Inscribed Angle Theorem The measure of an inscribed angle is exactly half the measure of its intercepted arc (or half the central angle that subtends the same arc). If $\angle ABC$ is an inscribed angle intercepting arc $AC$, then: $\angle ABC = \frac{1}{2} \times \text{arc } AC$

2. Angles Subtending the Same Arc If two or more inscribed angles intercept the exact same arc, then those angles are equal in measure.

3. Angles in a Semicircle An angle inscribed in a semicircle (meaning the angle intercepts the diameter of the circle) is always a right angle ($90^\circ$).

Cyclic Quadrilaterals

A cyclic quadrilateral is a four-sided polygon where all four of its vertices lie on the circumference of a single circle.

The most important property of a cyclic quadrilateral is that its opposite angles are supplementary, meaning they add up to $180^\circ$. For a cyclic quadrilateral $ABCD$: $\angle A + \angle C = 180^\circ$ $\angle B + \angle D = 180^\circ$

Example Problems

Example 1: Finding an Intercepted Arc If an inscribed angle $\angle ABC = 35^\circ$, find the measure of the intercepted arc $AC$.

Solution: According to the Inscribed Angle Theorem, the intercepted arc is twice the measure of the inscribed angle. $\text{arc } AC = 2 \times \angle ABC$ $\text{arc } AC = 2 \times 35^\circ = 70^\circ$

Example 2: Cyclic Quadrilateral Angles In cyclic quadrilateral $ABCD$, the measure of $\angle A = 80^\circ$. Find the measure of the opposite angle, $\angle C$.

Solution: Opposite angles in a cyclic quadrilateral are supplementary. $\angle A + \angle C = 180^\circ$ $80^\circ + \angle C = 180^\circ$ $\angle C = 180^\circ - 80^\circ = 100^\circ$