Lesson 2

Inscribed Angles

  • Let’s analyze angles made from chords.

2.1: Notice and Wonder: A New Angle

What do you notice? What do you wonder?

Circle A with angle BDC cut through it. Has arc BC.

2.2: A Central Relationship

Use the applet to answer the questions. Do not show the angle measures until you are told to.

  1. Name the central angle in this figure.
  2. Name the inscribed angle in this figure.
  3. Move point \(B\) around the circle. As you move this point, what happens to the measure of angle \(QBC\)? Show the angle measures to confirm.
  4. Move points \(C,Q,\) and \(B\) to new positions. Record the measure of angles \(QAC\) and \(QBC\). Repeat this several times.
  5. Make a conjecture about the relationship between an inscribed angle and the central angle that defines the same arc.

Here is a special case of an inscribed angle where one of the chords that defines the inscribed angle goes through the center. The central angle \(DCF\) measures \(\theta\) degrees, and the inscribed angle \(DEF\) measures \(\alpha\) degrees. Prove that \(\alpha=\frac12 \theta\).

A circle centered at C. The central angle  D C F measures theta degrees, and the inscribed angle D E F measures alpha degrees.


2.3: Similarity Returns

The image shows a circle with chords \(CD,CB,ED,\) and \(EB\). The highlighted arc from point \(C\) to point \(E\) measures 100 degrees. The highlighted arc from point \(D\) to point \(B\) measures 140 degrees.

Prove that triangles \(CFD\) and \(EFB\) are similar.

Circle with chords C D, C B, E D, and E B. E D and C B pass through F, forming triangles C F D and E F B. Arc C E and arc D B are highlighted.



We have discussed central angles such as angle \(AOB\). Another kind of angle in a circle is an inscribed angle, or an angle formed by 2 chords that share an endpoint. In the image, angle \(ACB\) is an inscribed angle.

Circle center O. Points A, B, C line on circle. Inscribed angles A O B and A O C.

It looks as though the inscribed angle is smaller than the central angle that defines the same arc. In fact, the measure of an inscribed angle is always exactly half the measure of the associated central angle. For example, if the central angle \(AOB\) measures 50 degrees, the inscribed angle \(ACB\) must measure 25 degrees, even if we move point \(C\) along the circumference (without going past \(A\) or \(B\)). This also means that all inscribed angles that define the same arc are congruent.

Glossary Entries

  • arc

    The part of a circle lying between two points on the circle.

  • central angle

    An angle formed by two rays whose endpoints are the center of a circle.

  • chord

    A chord of a circle is a line segment both of whose endpoints are on the circle.

  • inscribed angle

    An angle formed by two chords in a circle that share an endpoint.