Before you read

Answer first — see what you already know.

In a circular amphitheater, a spotlight at the center creates a $110°$ central angle. A spectator on the circle sees the same lit arc. What is the spectator's inscribed angle?

A
$55°$
B
$70°$
C
$110°$
D
$220°$

Check your answer

Answer:

Before you read

Answer first — see what you already know.

A triangle is inscribed in a circular clock face. One side of the triangle is a diameter of the clock. What is the angle opposite the diameter?

A
$45°$
B
$60°$
C
$90°$
D
It depends on the triangle

Check your answer

Answer:

Theory

Central and Inscribed Angles

Central angle: Vertex at the center of the circle. The central angle equals the arc it intercepts.

Inscribed angle: Vertex ON the circle. An inscribed angle is half the central angle that intercepts the same arc:
$\text{Inscribed angle} = \frac{1}{2} \times \text{Central angle}$

Inscribed angle = ½ × central angle

Special cases:
- An inscribed angle that intercepts a semicircle (diameter) is always $90°$ .
- Inscribed angles that intercept the same arc are equal.

Example 1

A central angle is $80°$ . What is the inscribed angle that intercepts the same arc?

Inscribed $= \frac{1}{2}(80°) = 40°$

40°

Strategy

SAT Pro Tip

The most tested fact: any inscribed angle subtending a diameter is exactly $90°$ . If you see a triangle inscribed in a circle with one side being a diameter, the angle opposite that side is a right angle.

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