Central and Inscribed Angles

SAT Math — Geometry & Trigonometry

In this lesson:

• Central angles equal their intercepted arcs
• Inscribed angles equal half the intercepted arc
• Inscribed angles on the same arc are congruent
• An inscribed angle on a diameter is always 90°

What You Will Learn Today

1. Central angle = intercepted arc
2. Inscribed angle = ½ × intercepted arc
3. Inscribed angles on the same arc are congruent
4. Inscribed angle on a diameter = 90°

Connecting to What You Know Already

From the previous lesson on arc measure:

• A full circle = 360° of arc
• Arc measure is in degrees, just like angles
• A chord connects two points on a circle

Key question: How does an angle relate to its arc — and does the vertex location matter?

Central Angles: Vertex at the Center

A central angle has its vertex at the center. Its sides are radii.

Rule: Central angle = intercepted arc (same measure)

Example:

• Minor arc
• Major arc

Inscribed Angle Is Half the Central Angle

The Inscribed Angle Theorem Explained

An inscribed angle has its vertex on the circle and its sides are chords.

Theorem: Inscribed angle = ½ × intercepted arc

Vertex location rule:

• Vertex at center → angle equals arc
• Vertex on circle → angle = half the arc

Example 1: Using the Inscribed Angle Theorem

Given: Arc . Find inscribed angle .

Given: Inscribed angle . Find arc .

The formula works in both directions.

Connecting Central and Inscribed Angles

Central angle . Point on the major arc. Find .

Step 1: Arc = (equals the central angle)

Step 2:

Same arc → inscribed angle = ½ × central angle

Quick Check: Central Versus Inscribed Angle

Central angle . Point on the major arc.

What is inscribed angle ?

Identify the intercepted arc, then apply the theorem.

Answer: Arc , so

Inscribed Angles Sharing the Same Arc

Corollary: Inscribed angles intercepting the same arc are congruent.

Each equals of the same arc, so they must be equal.

If , then — regardless of where and are on the major arc.

The Semicircle Theorem: Diameter Makes 90°

An inscribed angle subtending a diameter is always 90°.

Why: Diameter = arc of , so inscribed angle =

Converse: A 90° inscribed angle means the chord is a diameter.

SAT tip: scan every circle diagram for a diameter — mark that angle 90° immediately.

Right Triangle Inscribed in a Circle

Example 3: Finding the Diameter

, , . Find the diameter.

Step 1: → is a diameter (semicircle theorem converse)

Step 2:

Diameter = 10, Radius = 5

Quick Check: What Does 90° Tell You?

In a circle, inscribed angle .

What can you conclude about chord ?

Think before advancing...

Answer: A 90° inscribed angle means the subtended chord is a diameter. Chord is a diameter of the circle.

SAT Problem 1: Two-Step Chain

Central angle . Point on the major arc. Find .

Step 1: Arc = (equals central angle)

Step 2:

Chain: central angle → arc → inscribed angle

SAT Problem 2: Semicircle and Pythagorean Theorem

Diameter , . Find .

Step 1: on circle + is diameter →

Step 2:

5-12-13 Pythagorean triple

SAT Problem 3: Applying the Same-Arc Corollary

. Both and are inscribed angles. Find .

Identify the intercepted arcs before calculating.

Answer: Both angles intercept arc → same-arc corollary →

No calculation — arc recognition only.

Summary: The Four Key Rules

✓ Central angle = intercepted arc (vertex at center)
✓ Inscribed angle = ½ × intercepted arc (vertex on circle)
✓ Same-arc corollary: inscribed angles on the same arc are congruent
✓ Semicircle theorem: inscribed angle on a diameter = 90°; converse holds

Watch Out: Four Common Mistakes

Inscribed angle ≠ arc — it equals half the arc
Intercepted arc is opposite the vertex — trace both sides to find it
Same-arc corollary: only applies when angles intercept the same arc
Diameter in a diagram → mark the inscribed angle 90° immediately

Coming Up: Tangent Lines and Chord Properties

Next lesson covers:

• Tangent-radius perpendicularity
• Tangent segments from an external point
• Chord intersection and segment relationships

These properties combine with today's inscribed angle rules in multi-step SAT problems.