The Inscribed Angle Theorem | How to Calculate and Apply Inscribed Angles in Circle Geometry

Inscribed Angle

An inscribed angle is an angle that is formed by two chords or arcs inside a circle

An inscribed angle is an angle that is formed by two chords or arcs inside a circle. The vertex of the angle is located on the circumference of the circle, and the sides of the angle are the chords or arcs that intersect at the vertex.

The main property of an inscribed angle is that it is equal in measure to half the measure of the arc it intercepts. This is known as the Inscribed Angle Theorem. In other words, if you take the measure of the arc (in degrees or radians) and divide it by 2, you will get the measure of the inscribed angle.

Inscribed angles have several useful properties and applications:

1. Central angles: The central angle that subtends the same arc as an inscribed angle has twice the measure of that inscribed angle. For example, if an inscribed angle has a measure of 30 degrees, then the central angle that subtends the same arc will have a measure of 60 degrees.

2. Intercepted arcs: The intercepted arc is the portion of the circle that lies between the two sides of the inscribed angle. The measure of the intercepted arc is twice the measure of the inscribed angle. For example, if an inscribed angle has a measure of 40 degrees, then the intercepted arc will have a measure of 80 degrees.

3. Tangent properties: If a tangent line intersects a chord or an inscribed angle, the angle between the tangent line and the chord or inscribed angle is equal to the measure of the inscribed angle. This is known as the Tangent-Chord Angle Theorem.

Inscribed angles are commonly used in geometry problems involving circles, such as determining the measure of missing angles or proving congruence between different angles. They provide a powerful tool for analyzing and solving circle-related problems.

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