Is an inscribed angle congruent to the arc?
Yes, an inscribed angle in a circle is congruent to the arc it intercepts. This is a fundamental geometric relationship known as the "Inscribed Angle Theorem." Here's what the theorem states:
- Inscribed Angle Theorem: An inscribed angle in a circle is congruent to half the measure of the central angle that subtends the same arc.
In simpler terms, if you have an inscribed angle that intercepts a particular arc in a circle, the measure of that inscribed angle is equal to half the measure of the central angle that covers the same arc.
Mathematically, if:
- θ represents the measure of the inscribed angle,
- α represents the measure of the intercepted arc, and
- Φ represents the measure of the central angle,
then θ = 0.5 * Φ, and θ = α/2.
This relationship is useful for solving various geometry problems involving circles and angles. It's important to remember that this theorem applies specifically to inscribed angles in circles and their relationships with the corresponding arcs.
