Inscribed Angles- Theorems and Examples

What Is an Inscribed Angle?

An inscribed angle is an angle formed by two chords in a circle that share an endpoint on the circle itself. That shared endpoint is the vertex of the angle, and it sits on the circle, not inside or outside it.

Think of it this way: if you pick any point on a circle's edge and draw two line segments from that point to two other points on the circle, you've created an inscribed angle. The arc between those two other points is called the intercepted arc.

The Inscribed Angle Theorem

Here's the part most geometry students miss: an inscribed angle is always half the measure of its intercepted arc. That's it. That's the whole theorem.

If you know the arc measure, you know the angle. If you know the angle, you multiply by 2 to get the arc.

Formula:

• Inscribed Angle = ½ × Intercepted Arc
• Intercepted Arc = 2 × Inscribed Angle

Inscribed Angle vs. Central Angle

A central angle has its vertex at the circle's center. An inscribed angle has its vertex on the circle. The central angle "sees" the same arc but measures twice as much.

Special Case: Angles Inscribed in a Semicircle

When an inscribed angle intercepts a diameter, something useful happens. The intercepted arc is 180°. Half of 180° is 90°.

Any angle inscribed in a semicircle is a right angle. This is Thales' theorem, and it's one of the oldest results in geometry.

No matter where you place the vertex on the semicircle arc, the angle stays exactly 90°.

Angles Subtending the Same Arc

All inscribed angles that intercept the same arc are equal. This holds true whether the angles are on the same side of the chord or opposite sides.

If ∠A and ∠B both intercept arc CD, then ∠A = ∠B. Simple as that.

Inscribed Angles and Quadrilaterals

When a quadrilateral is inscribed in a circle (called a cyclic quadrilateral), opposite angles add up to 180°. This comes directly from the inscribed angle theorem.

Angle A + Angle C = 180°

Angle B + Angle D = 180°

How to Solve Inscribed Angle Problems

Step 1: Identify the intercepted arc

Find the two points where the angle's rays hit the circle. The arc between those points is your intercepted arc.

Step 2: Find the arc measure

Use given information. If the problem gives you a central angle, that's the arc measure. If it gives you another inscribed angle intercepting the same arc, those angles are equal.

Step 3: Apply the formula

Divide the arc measure by 2 to get the inscribed angle, or multiply the inscribed angle by 2 to get the arc.

Example Problem

Problem: Circle O has points A, B, and C on its circumference. If central angle AOC measures 80°, what is the measure of inscribed angle ABC?

Solution: Arc AC = 80° (central angle equals arc). Angle ABC intercepts arc AC. Inscribed angle = ½ × 80° = 40°

Common Mistakes to Avoid

• Confusing the inscribed angle with the central angle that sees the same arc
• Forgetting that inscribed angles can never exceed 180°
• Not checking which arc is being intercepted when angles face both directions

Quick Reference Table

Master these relationships and inscribed angle problems become straightforward. The theorem does the heavy lifting—you just need to identify which arc you're working with. 🔺