Inscribed Angle Theorem

Definition

The Inscribed Angle Theorem states that an inscribed angle in a circle is half the measure of the intercepted arc. This relationship highlights how angles and arcs interact within a circle, making it crucial for solving various geometric problems involving circles. Understanding this theorem allows for deeper insights into the properties of circles, including relationships between angles, arcs, and chords.

5 Must Know Facts For Your Next Test

1. The measure of an inscribed angle is always half the measure of its intercepted arc, which can be written as $$m\angle = \frac{1}{2} m(arc)$$.
2. If two inscribed angles intercept the same arc, they are congruent, meaning they have the same measure.
3. An inscribed angle that intercepts a semicircle (an arc measuring 180 degrees) is a right angle.
4. The Inscribed Angle Theorem can be applied to solve problems involving cyclic quadrilaterals, where opposite angles are supplementary.
5. Inscribed angles can be used to derive relationships between chords and arcs in more complex geometric scenarios.

Review Questions

• How does the Inscribed Angle Theorem relate to the measures of central angles and intercepted arcs in a circle?
  • The Inscribed Angle Theorem establishes a direct relationship between inscribed angles, central angles, and intercepted arcs. Specifically, while an inscribed angle measures half the intercepted arc, a central angle that subtends the same arc will equal the measure of that arc. This connection helps in solving geometric problems where you need to relate various types of angles within a circle.
• Explain how the properties of inscribed angles can be used to solve problems involving cyclic quadrilaterals.
  • In cyclic quadrilaterals, opposite angles are supplementary due to the properties of inscribed angles. This means if one angle measures $$x$$ degrees, then its opposite angle will measure $$180 - x$$ degrees. By applying the Inscribed Angle Theorem, we can find unknown angles by using this property and the measures of intercepted arcs created by diagonals or sides of the quadrilateral.
• Evaluate a situation where two inscribed angles intercept the same arc. What conclusions can you draw about their measures?
  • When two inscribed angles intercept the same arc, we can conclude that those angles are congruent. This is a direct application of the Inscribed Angle Theorem, which asserts that each angle measures half of the intercepted arc's degree measure. Thus, regardless of their positions on the circle, as long as they intercept the same arc, their measures will be identical.

Related terms

Central Angle: An angle whose vertex is at the center of the circle and whose sides intersect the circle.

Intercepted Arc: The arc that lies in the interior of an inscribed angle and has endpoints on the angle's sides.

Chord:

A line segment with both endpoints on the circle, which can help define angles and arcs within the circle.