Inscribed Angle Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Inscribed Angle.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An angle whose vertex lies on the circle and whose sides are chords of the circle. Its measure is exactly half the measure of the intercepted arc.

Imagine sitting on the edge of a circular stadium and looking at two players on the field. The angle your eyes make is an inscribed angle. No matter where you sit on the same arc, that viewing angle stays the same—and it's always half of what you'd see from the center. It's like the circle is 'halving' your perspective compared to the center's view.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An inscribed angle is always half the central angle that intercepts the same arc.

Common stuck point: An angle inscribed in a semicircle (intercepting a 180° arc) is always 90°. This is called Thales' theorem.

Worked Examples

Example 1

easy
An inscribed angle intercepts an arc of 80°. What is the measure of the inscribed angle?

Solution

  1. 1
    Step 1: Recall the Inscribed Angle Theorem: an inscribed angle equals half the intercepted arc. That is, \angle = \frac{1}{2} \times \text{arc}.
  2. 2
    Step 2: Substitute the intercepted arc measure: \angle = \frac{1}{2} \times 80°.
  3. 3
    Step 3: Compute the result: \angle = 40°.

Answer

40°
The Inscribed Angle Theorem states that an inscribed angle is exactly half the measure of its intercepted arc. Here, half of 80° gives 40°.

Example 2

medium
In circle O, inscribed angle \angle ABC intercepts arc AC. If arc AC = 134°, and arc CD = 70°, find inscribed angle \angle ADC that intercepts arc AC from the same side.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
An inscribed angle measures 35°. What is the measure of its intercepted arc?

Example 2

hard
Quadrilateral ABCD is inscribed in a circle. If \angle A = 82°, find \angle C. Then, if arc AB = 96° and arc BC = 110°, find \angle ADC.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

central angle