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Geometry

Inscribed Angles and Circle Theorems

Lesson

An inscribed anglehas its vertex ON the circle and its sides as chords. It’s half the central angle subtending the same arc.

Inscribed angle theorem

inscribed angle=12⋅intercepted arc\text{inscribed angle} = \tfrac{1}{2} \cdot \text{intercepted arc}

Inscribed in a semicircle

Any angle inscribed in a semicircle is 90∘90^\circ. (The arc is 180°; half is 90°.)

Cyclic quadrilateral

A quadrilateral whose four vertices lie on a circle has opposite angles summing to 180°.

Tangent-chord angle

Angle between a tangent and a chord drawn from the point of tangency equals half the intercepted arc.

Worked example

Inscribed angle intercepts arc 80°: angle is 80/2=40∘80/2 = 40^\circ.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

Inscribed angle, arc 80∘\text{Inscribed angle, arc } 80^\circ

Problem 2

Inscribed angle, arc 100∘\text{Inscribed angle, arc } 100^\circ

Problem 3

Angle inscribed in a semicircle\text{Angle inscribed in a semicircle}

Problem 4

Inscribed angle 30∘. Arc?\text{Inscribed angle } 30^\circ. \text{ Arc?}

Practice

Standard problems matching the lesson.

Problem 5

Inscribed angle 45∘. Arc?\text{Inscribed angle } 45^\circ. \text{ Arc?}

Problem 6

Arc 120∘. Inscribed angle?\text{Arc } 120^\circ. \text{ Inscribed angle?}

Problem 7

Angle in semicircle\text{Angle in semicircle}

Problem 8

Inscribed angle for arc 60∘\text{Inscribed angle for arc } 60^\circ

Problem 9

Inscribed angle 25∘. Arc?\text{Inscribed angle } 25^\circ. \text{ Arc?}

Problem 10

Tangent-chord angle 35∘. Arc?\text{Tangent-chord angle } 35^\circ. \text{ Arc?}

Problem 11

Tangent-chord intercepts arc 140∘. Angle?\text{Tangent-chord intercepts arc } 140^\circ. \text{ Angle?}

Problem 12

An angle inscribed in semicircle is always?\text{An angle inscribed in semicircle is always?}

Problem 13

Arc 150∘. Inscribed?\text{Arc } 150^\circ. \text{ Inscribed?}

Problem 14

Arc 36∘. Inscribed?\text{Arc } 36^\circ. \text{ Inscribed?}

Problem 15

Two inscribed angles subtending arc 80∘. Each?\text{Two inscribed angles subtending arc } 80^\circ. \text{ Each?}

Problem 16

Inscribed quadrilateral: opposite angles sum to?\text{Inscribed quadrilateral: opposite angles sum to?}

Problem 17

Cyclic quad has angle 70∘. Opposite angle?\text{Cyclic quad has angle } 70^\circ. \text{ Opposite angle?}

Problem 18

Cyclic quad has angle 95∘. Opposite?\text{Cyclic quad has angle } 95^\circ. \text{ Opposite?}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

Arc 240∘. Inscribed angle?\text{Arc } 240^\circ. \text{ Inscribed angle?}

Problem 20

Arc 50∘. Inscribed angle?\text{Arc } 50^\circ. \text{ Inscribed angle?}

Problem 21

Cyclic quad angle 110∘. Opposite?\text{Cyclic quad angle } 110^\circ. \text{ Opposite?}

Problem 22

Tangent-chord angle 45∘. Arc?\text{Tangent-chord angle } 45^\circ. \text{ Arc?}

Problem 23

A right angle is inscribed. Arc measure?\text{A right angle is inscribed. Arc measure?}

Problem 24

Inscribed angles intercept arcs 60∘ and 80∘. Sum of angles?\text{Inscribed angles intercept arcs } 60^\circ \text{ and } 80^\circ. \text{ Sum of angles?}

Problem 25

Tangent-tangent angle from external point, arcs 120∘ and 240∘. Angle?\text{Tangent-tangent angle from external point, arcs } 120^\circ \text{ and } 240^\circ. \text{ Angle?}

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