# 15.1 Central Angles and Inscribed Angles

Central angles and inscribed angles are two important concepts in geometry. A central angle is an angle whose vertex is the center of a circle. An inscribed angle is an angle whose vertex lies on a circle and whose sides contain chords of the circle. There are many properties of central angles and inscribed angles that make them useful for solving geometry problems. In this article, we will discuss the properties of central angles and inscribed angles and how they can be used to solve geometry problems.

## Properties of Central Angles The most important property of a central angle is that its measure is always equal to the measure of the arc it subtends. A central angle is defined as an angle whose vertex is the center of a circle; therefore, the measure of the angle is equal to the measure of the arc that it subtends. This is true regardless of the size of the angle or the size of the circle.

In addition, a central angle is also equal to half of the measure of the intercepted arc. An intercepted arc is an arc that lies between the sides of an angle and is created when a central angle is drawn. The measure of the intercepted arc is equal to half the measure of the central angle. This property can be used to solve geometry problems involving the measure of a central angle and the measure of the intercepted arc.

## Properties of Inscribed Angles Inscribed angles have several properties that are useful for solving geometry problems. The most important property is that the measure of an inscribed angle is always half the measure of the intercepted arc. This property is true regardless of the size of the angle or the size of the circle. Inscribed angles also have the property that the measure of the intercepted arc is equal to the sum of the measures of the two chords that create the inscribed angle.

In addition, inscribed angles also have the property that the measure of the inscribed angle is equal to the measure of the central angle that subtends the same arc. This property can be used to solve geometry problems involving the measure of an inscribed angle and the measure of a central angle.

## Using Central Angles and Inscribed Angles to Solve Geometry Problems Central angles and inscribed angles can be used to solve geometry problems. By knowing the properties of central angles and inscribed angles, it is possible to solve problems involving the measure of arcs, angles, and chords. The properties discussed in this article can be used to find the measure of a central angle or an inscribed angle given the measure of the arc or a chord.

In addition, central angles and inscribed angles can be used to find the measure of an arc or a chord given the measure of a central angle or an inscribed angle. By knowing the properties of central angles and inscribed angles, it is possible to find the measure of any arc or chord given the measure of any angle.

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Central angles and inscribed angles are two important concepts in geometry. Knowing the properties of central angles and inscribed angles can be useful for solving geometry problems involving the measure of arcs, angles, and chords. By understanding the properties of central angles and inscribed angles, it is possible to find the measure of any arc or chord given the measure of any angle.