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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 \cdot 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:

$A S = B S = P S$ , as they are all radii of the circle.
Thus, $△ A S P$ and $△ B S P$ both have two equally sized legs.
This means that $w = 180^{\circ} - 2 x$ ,
and $z = 180^{\circ} - 2 y$ .
Thus, $u = 360^{\circ} - w - z$ .
You also know that $v = x + y$ .

When combining all this information, you get:

\begin{array}{l} u & = 360^{\circ} - w - z \\ = 360^{\circ} - (180^{\circ} - 2 x) - (180^{\circ} - 2 y) \\ = 360^{\circ} - 180^{\circ} + 2 x - 180^{\circ} + 2 y \\ = 2 x + 2 y \\ = x + x + y + y \\ = (x + y) + (x + y) \\ = v + v \\ = 2 v \end{array}

Q.E.D

Example 1

Find all the angles in the following triangles:

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

∠ S = 2 \cdot 40^{\circ} = 80^{\circ}

Triangle $△ A S C$ has two equal sides as $A S = C S$ —they are both radiuses of the circle. Thus

∠ C A S = ∠ A C S = \frac{180^{\circ} - 80^{\circ}}{2} = 50^{\circ}

Triangle $△ A B C$ has an angle $∠ B = 40^{\circ}$ ; you need to find angles $∠ C A B$ and $∠ A C B$ . You already know that angle $∠ S A B = 20^{\circ}$ , so you get

∠ C A B = 50^{\circ} + 20^{\circ} = 70^{\circ}

The final angle is then

∠ A C B = 180^{\circ} - 40^{\circ} - 70^{\circ} = 70^{\circ}

You have now found all the angles in both triangles.

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Application of Opposite Angles

Constructing a 60°, 30° or 15° Angle

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