Inscribed Angle Examples: 25 Problems with Answers

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°. What is the measure of the inscribed angle?

Answer

40°

First step

1
Step 1: Recall the Inscribed Angle Theorem: an inscribed angle equals half the intercepted arc. That is, ∠=12×arc.

Full solution

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

Example 3

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

Example 4

medium
In circle O, points A, B, C, D lie on the circle. Arcs: AB=80°, BC=100°, CD=60°, DA=120° (totals 360°). Find inscribed angle ∠BAD.

Example 5

medium
In circle O, BD is a diameter and A is on the circle. If arc AB=130°, find ∠ADB.

Example 6

hard
In circle O, two chords AB and CD intersect inside at P. Arc AC=84° and arc BD=36°. Find ∠APC.

Example 7

hard
In circle O, A, B, C, D lie in order on the circle. Inscribed angle ∠BAC=28° and ∠CAD=47°. Find arc BD (not containing A).

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 ∠A=82°, find ∠C. Then, if arc AB=96° and arc BC=110°, find ∠ADC.

Example 3

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

Example 4

easy
In a circle, a central angle measures 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°. What does the other measure?

Example 6

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

Example 7

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

Example 8

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

Example 9

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

Example 10

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

Example 11

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

Example 12

hard
In circle O, chord AB subtends an arc of 140° on one side and 220° on the other. Points C and D are on opposite arcs. Find ∠ACB and ∠ADB.

Example 13

hard
In circle O, ∠ABC=40° is inscribed and intercepts arc AC. Chord AC is extended to meet chord BD outside the circle at P, where arc BD (not containing C) is 30°. Find arcs AC and the angle at P.

Example 14

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

Example 15

hard
Cyclic quadrilateral ABCD has ∠A=(2x+10)° and ∠C=(3x−5)°. Find x and ∠A.

Example 16

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

Example 17

challenge
In cyclic quadrilateral ABCD, the diagonals AC and BD meet at E. Arcs AB=70°, BC=80°, CD=100°, DA=110°. Find ∠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