Inscribed angle = half of the central angle

Vocabulary

• Central angle: angle whose vertex is at the center of the circle.
• Inscribed angle: angle whose vertex is on the circle and sides are chords.
• Both intercept the same arc if their sides pass through the same points of the circle.

The theorem

If an inscribed angle and a central angle intercept the same arc:

$\widehat{\text{central angle}} = 2 \times \widehat{\text{inscribed angle}}$

Equivalently:

$\widehat{\text{inscribed angle}} = \dfrac{1}{2} \times \widehat{\text{central angle}}$

Example

If the central angle is $80°$, the inscribed angle intercepting the same arc is $40°$.

If the inscribed angle is $30°$, the corresponding central angle is $60°$.

Corollary 1: inscribed angles on the same arc

Two inscribed angles intercepting the same arc are equal.

Corollary 2: inscribed angle in a semicircle

If one side of the inscribed angle is a diameter, the central angle is $180°$, so the inscribed angle is $90°$.

In other words: if a triangle has one side on the diameter of the circumscribed circle, it is right-angled (at the opposite vertex).

Method: finding a missing angle

1. Identify the intercepted arc.
2. Find a central angle or another inscribed angle intercepting the same arc.
3. Apply $\times 2$ or $\div 2$.

Common mistakes

1. Confusing central and inscribed angles (check where the vertex is).
2. Angles not intercepting the same arc (essential!).
3. Applying $\times 2$ the wrong way: the CENTER is the double.

?À toi de jouer

A central angle measures 120°. The inscribed angle intercepting the same arc?

A.60°B.120°C.240°D.90°

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