What is an angle whose vertex is on a circle and whose sides contain chords of the circle?

General
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what is an angle whose vertex is on a circle and whose sides contain chords of the circle?

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What is an angle whose vertex is on a circle and whose sides contain chords of the circle?

Answer:
An angle whose vertex is on a circle and whose sides contain chords of the circle is called an inscribed angle.

Detailed Explanation:

1. Definition of Inscribed Angle:
An inscribed angle is formed when two chords in a circle share an endpoint. This common endpoint is the vertex of the angle, and the other two endpoints lie on the circle, thereby creating two intersecting chords. The measure of an inscribed angle is directly related to the arc it intercepts.

2. Intercepted Arc:
The arc that lies inside the inscribed angle and whose endpoints are the endpoints of the angle’s sides is known as the intercepted arc.

Relationship Between Inscribed Angle and Intercepted Arc:

The Measure of an Inscribed Angle:

  • The measure of an inscribed angle is always half the measure of its intercepted arc.

If we denote the measure of the inscribed angle as \theta and the measure of the intercepted arc as m, the relationship can be mathematically expressed as:

\theta = \frac{m}{2}

Solution By Steps:

  1. Identify the Vertex and Chords:

    • The vertex of the inscribed angle lies on the circumference of the circle.
    • The sides of the angle are formed by two chords of the circle.

  2. Identify the Intercepted Arc:

    • The intercepted arc is the portion of the circle that lies “inside” the inscribed angle, between the endpoints of the chords.

  3. Calculate the Angle:

    • Measure the intercepted arc.
    • Divide the measure of the intercepted arc by 2 to find the measure of the inscribed angle.

Example:

Consider a circle with an inscribed angle A\hat{B}C where B is the vertex on the circle, and A and C are points on the circle forming chords AB and BC. Suppose the intercepted arc AC has a measure of 120^\circ.

  1. Intercepted Arc Calculation:

    • Measure of the intercepted arc AC: 120^\circ.

  2. Inscribed Angle Calculation:

    • Measure of the inscribed angle A\hat{B}C: \theta = \frac{120^\circ}{2} = 60^\circ.

Therefore, \theta = 60^\circ.

Final Answer:
The angle whose vertex is on a circle and whose sides contain chords of the circle is called an inscribed angle. The measure of an inscribed angle is half the measure of its intercepted arc.