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Inscribed Angles

Study Guide

Key Definition

An inscribed angle in a circle is formed by two chords that have a common endpoint on the circle. This common endpoint is the vertex of the angle. The measure of an inscribed angle is half the measure of its intercepted arc: $m\angle ABC = \frac{1}{2} m\widehat{AC}$ (where m\widehat{AC} = m\angle AOC).

Important Notes

  • Inscribed angles that intercept the same arc are congruent.
  • $\angle ADC \cong \angle ABC \cong \angle AFC$ (each intercepts arc AC; see Figure 1).
  • The central angle is twice any inscribed angle intercepting the same arc.
  • In a semicircle, the inscribed angle is a right angle.

Mathematical Notation

$\angle$ represents an angle$\cong$ means congruent$\frac{1}{2}$ represents one halfRemember to use proper notation when solving problems

Why It Works

The Inscribed Angle Theorem works because the central angle is always twice the inscribed angle for the same arc, a property derived from the circle's geometry.

Remember

An inscribed angle's measure is always half of its intercepted arc's measure.

Quick Reference

Inscribed Angle Theorem:$m\angle ABC = \frac{1}{2} m\widehat{AC}$ (where m\widehat{AC} = m\angle AOC)

Understanding Inscribed Angles

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Video explanation of this concept

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Beginner Explanation

Inscribed angles are angles made by two chords in a circle that have the same endpoint. They are half the measure of the intercepted arc.Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

In a circle with center O, let A and C be endpoints of an arc and B a point on the circle such that $\angle AOC$ is the central angle and $\angle ABC$ the corresponding inscribed angle. If $m\angle AOC = 80^\circ$, what is $m\angle ABC$?

A$40^\circ$B$80^\circ$C$160^\circ$D$20^\circ$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

Imagine a pizza slice forming an inscribed angle in a circular pizza. If the arc of the pizza slice measures $150^\circ$, what is the angle of the slice?Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

Consider a circle where two inscribed angles intercept the same arc. How can you prove these angles are congruent?

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

In a circle with center O, let D and F be endpoints of arc DF, and E and G points on the circle such that $\angle DEF$ and $\angle DGF$ intercept arc DF. If $\angle DEF = 70^\circ$ and $\angle DGF = 70^\circ$, what is the measure of arc DF?

A$120^\circ$B$130^\circ$C$140^\circ$D$150^\circ$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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