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How to Prove the Inscribed Angle Theorem

Circle with an inscribed angle and a central angle

An angle which sits on the boundary of a circle is called an inscribed angle.

An angle which sits in the center of a circle is called a central angle.

Formula

Inscribed Angle Theorem

If the inscribed angle v spans the same circular sector as the central angle u, then the central angle is twice the size of the inscribed angle:

u = 2 ⋅ v

Think About This

Proof of the Inscribed Angle Theorem

Look closely at the figure below!

Proof of inscribed angle theorem

From the figure you get the following information:

  • AS = BS = PS, as they are all radii of the circle.

  • Thus, △ASP and △BSP both have two equally sized legs.

  • This means that w = 180∘ − 2x,

  • and z = 180∘ − 2y.

  • Thus, u = 360∘ − w − z.

  • You also know that v = x + y.

When combining all this information, you get:

u = 360∘ − w − z = 360∘ − (180∘ − 2x) − (180∘ − 2y) = 360∘ − 180∘ + 2x − 180∘ + 2y = 2x + 2y = x + x + y + y = (x + y) + (x + y) = v + v = 2v

Q.E.D

Example 1

Find all the angles in the following triangles:

Example of innscribed angle and central angle

When you know the inscribed angle, you can find the central angle, because it is always exactly two times larger. Thus

∠S = 2 ⋅ 40∘ = 80∘

Triangle △ASC has two equal sides as AS = CS—they are both radiuses of the circle. Thus

∠CAS = ∠ACS = 180∘ − 80∘ 2 = 50∘

Triangle △ABC has an angle ∠B = 40∘; you need to find angles ∠CAB and ∠ACB. You already know that angle ∠SAB = 20∘, so you get

∠CAB = 50∘ + 20∘ = 70∘

The final angle is then

∠ACB = 180∘ − 40∘ − 70∘ = 70∘

You have now found all the angles in both triangles.

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