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.

Read the full concept explanation →

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 angle with its vertex on the circle measures half the arc it intercepts.

Common stuck point: The procedure for inscribed angle is the easy part; the trap is setting the inscribed angle equal to the arc. Asking "Is the angle's vertex on the circle (not the center), with both sides being chords?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the angle's vertex on the circle (not the center), with both sides being chords?

Worked Examples

Example 1

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

Answer

40°40°

First step

1
Step 1: Recall the Inscribed Angle Theorem: an inscribed angle equals half the intercepted arc. That is, ∠=12×arc\angle = \frac{1}{2} \times \text{arc}.

Full solution

  1. 2
    Step 2: Substitute the intercepted arc measure: ∠=12×80°\angle = \frac{1}{2} \times 80°.
  2. 3
    Step 3: Compute the result: ∠=40°\angle = 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 OO, inscribed angle ∠ABC\angle ABC intercepts arc ACAC. If arc AC=134°AC = 134°, and arc CD=70°CD = 70°, find inscribed angle ∠ADC\angle ADC that intercepts arc ACAC from the same side.

Example 3

easy
Arc ABAB in circle OO measures 96°96°. Inscribed angle ∠ACB\angle ACB has its vertex on the major arc. Find ∠ACB\angle ACB.

Example 4

medium
In circle OO, points AA, BB, CC, DD lie on the circle. Arcs: AB=80°AB = 80°, BC=100°BC = 100°, CD=60°CD = 60°, DA=120°DA = 120° (totals 360°360°). Find inscribed angle ∠BAD\angle BAD.

Example 5

medium
In circle OO, BDBD is a diameter and AA is on the circle. If arc AB=130°AB = 130°, find ∠ADB\angle ADB.

Example 6

hard
In circle OO, two chords ABAB and CDCD intersect inside at PP. Arc AC=84°AC = 84° and arc BD=36°BD = 36°. Find ∠APC\angle APC.

Example 7

hard
In circle OO, AA, BB, CC, DD lie in order on the circle. Inscribed angle ∠BAC=28°\angle BAC = 28° and ∠CAD=47°\angle CAD = 47°. Find arc BDBD (not containing AA).

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°35°. What is the measure of its intercepted arc?

Example 2

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

Example 3

easy
An inscribed angle intercepts an arc of 60°60°. Find the inscribed angle.

Example 4

easy
In a circle, a central angle measures 140°140°. An inscribed angle intercepts the same arc. Find the inscribed angle.

Example 5

easy
Two inscribed angles in a circle intercept the same arc. One angle measures 42°42°. What does the other measure?

Example 6

medium
Quadrilateral PQRSPQRS is inscribed in a circle. If ∠P=110°\angle P = 110°, find ∠R\angle R.

Example 7

medium
In circle OO, inscribed angle ∠ABC=(3x+5)°\angle ABC = (3x+5)° intercepts arc AC=(8x−10)°AC = (8x-10)°. Find xx.

Example 8

medium
A triangle is inscribed in a circle. Its arcs are 80°80°, 130°130°, and 150°150°. Find the three angles of the triangle.

Example 9

medium
In cyclic quadrilateral ABCDABCD, ∠B=95°\angle B = 95° and ∠A=70°\angle A = 70°. Find ∠C\angle C and ∠D\angle D.

Example 10

medium
In circle OO, arc AB=4x°AB = 4x°, and inscribed angle ∠ACB=(x+30)°\angle ACB = (x+30)°. Find xx.

Example 11

medium
Triangle ABCABC is inscribed in a circle with ABAB as a diameter. If ∠ABC=35°\angle ABC = 35°, find ∠BAC\angle BAC.

Example 12

hard
In circle OO, chord ABAB subtends an arc of 140°140° on one side and 220°220° on the other. Points CC and DD are on opposite arcs. Find ∠ACB\angle ACB and ∠ADB\angle ADB.

Example 13

hard
In circle OO, ∠ABC=40°\angle ABC = 40° is inscribed and intercepts arc ACAC. Chord ACAC is extended to meet chord BDBD outside the circle at PP, where arc BDBD (not containing CC) is 30°30°. Find arcs ACAC and the angle at PP.

Example 14

hard
A tangent and a chord meet at point TT on a circle. The chord cuts an arc of 110°110° on the near side. Find the tangent-chord angle.

Example 15

hard
Cyclic quadrilateral ABCDABCD has ∠A=(2x+10)°\angle A = (2x+10)° and ∠C=(3x−5)°\angle C = (3x-5)°. Find xx and ∠A\angle A.

Example 16

hard
Two secants from external point PP cut arcs 150°150° (far) and 50°50° (near). Find ∠P\angle P.

Example 17

challenge
In cyclic quadrilateral ABCDABCD, the diagonals ACAC and BDBD meet at EE. Arcs AB=70°AB = 70°, BC=80°BC = 80°, CD=100°CD = 100°, DA=110°DA = 110°. Find ∠AEB\angle AEB.

Example 18

challenge
A regular pentagon is inscribed in a circle. Find the measure of one inscribed angle subtended by two adjacent vertices from a third vertex on the major arc.
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Background Knowledge

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

central angle