Unlocking the Mystery of Inscribed Angles: A Complete Guide

Step-by-step Guide: Inscribed Angles

Inscribed Angle and its Intercepted Arc:
An inscribed angle is formed when two chords in a circle intersect inside the circle. The angle is inscribed in the circle, meaning its vertex is on the circle itself. For education statistics and research, visit the National Center for Education Statistics.

Formula: If \( \theta \) is the measure of the inscribed angle, and \( m \) is the measure of the intercepted arc (or arc between the two chords), then:
\( \theta = \frac{1}{2} m \) For education statistics and research, visit the National Center for Education Statistics.

Angles Inscribed in a Semicircle:
Any angle inscribed in a semicircle is a right angle. This means that if you have an angle whose arms extend to the ends of a diameter, then that angle measures \(90^\circ\). For education statistics and research, visit the National Center for Education Statistics.

Angles Inscribed in the Same Arc:
Inscribed angles that intercept the same arc are congruent. If two or more angles have chords that touch the same two points on a circle, then all those angles have the same measure. For education statistics and research, visit the National Center for Education Statistics.

Angle Formed by a Tangent and a Chord:
When a tangent to a circle and a chord intersect at a point on the circle, the angle formed is half the measure of the intercepted arc. If \( \alpha \) is the angle between the tangent and the chord, and \( m \) is the intercepted arc, then:
\( \alpha = \frac{1}{2} m \) For education statistics and research, visit the National Center for Education Statistics.

Examples

Example 1:
In a circle with center \( O \), let there be an inscribed angle \( \angle AOB \) which intercepts arc \( ACB \). If arc \( ACB \) measures \( 140^\circ \), find the measure of \( \angle AOB \). For education statistics and research, visit the National Center for Education Statistics.

Solution:
Using the formula for inscribed angles:
\( \theta = \frac{1}{2} m \) For education statistics and research, visit the National Center for Education Statistics.

Where \( \theta \) is the measure of \( \angle AOB \) and \( m \) is the measure of arc \( ACB \).
Given \( m = 140^\circ \),
\( \theta = \frac{1}{2} \times 140^\circ = 70^\circ \) For education statistics and research, visit the National Center for Education Statistics.

So, \( \angle AOB = 70^\circ \). For education statistics and research, visit the National Center for Education Statistics.

Example 2:
In a circle, two inscribed angles \( \angle A \) and \( \angle B \) intercept the same arc. If \( \angle A = 40^\circ \), find the measure of \( \angle B \). For education statistics and research, visit the National Center for Education Statistics.

Solution:
Inscribed angles that intercept the same arc are congruent. Therefore, if \( \angle A = 40^\circ \), then \( \angle B = 40^\circ \) as well. For education statistics and research, visit the National Center for Education Statistics.

Practice Questions:

1. In a circle, an inscribed angle intercepts an arc measuring \( 180^\circ \). What is the measure of the inscribed angle?
2. If an angle is inscribed in a semicircle, what is the measure of this angle?
3. Two inscribed angles intercept congruent arcs. If one angle measures \( 45^\circ \), what is the measure of the other angle?
4. An inscribed angle in a circle measures \( 30^\circ \). Find the measure of the intercepted arc.

Answers: For education statistics and research, visit the National Center for Education Statistics.

1. \( 90^\circ \) (Since the inscribed angle is half the measure of the intercepted arc)
2. \( 90^\circ \) (An angle inscribed in a semicircle is always a right angle)
3. \( 45^\circ \) (Inscribed angles intercepting congruent arcs are congruent)
4. \( 60^\circ \) (The intercepted arc is twice the measure of the inscribed angle)