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.

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:
\( \theta = \frac{1}{2} m \)

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 \(90^\circ\).

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.

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:
\( \alpha = \frac{1}{2} m \)

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 \( \angle AOB \).

Solution:
Using the formula for inscribed angles:
\( \theta = \frac{1}{2} m \)

Where \( \theta \) is the measure of \( \angle AOB \) and \( m \) is the measure of arc \( ACB \).
Given \( m = 140^\circ \),
\( \theta = \frac{1}{2} \times 140^\circ = 70^\circ \)

So, \( \angle AOB = 70^\circ \).

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 \( \angle B \).

Solution:
Inscribed angles that intercept the same arc are congruent. Therefore, if \( \angle A = 40^\circ \), then \( \angle B = 40^\circ \) as well.

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:

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)