Central and Inscribed Angles

SAT Math — Geometry & Trigonometry

In this lesson:

• An arc's measure is its central angle — not its length
• 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. Find angles, arc measures, and lengths; use isosceles and right triangles
2. State an arc's measure from its central angle, not its length
3. Inscribed angle = ½ × intercepted arc
4. Inscribed angles on the same arc are congruent
5. Inscribed angle in a semicircle = 90°

Connecting to What You Know Already

Ten-second check: a central angle cuts off what fraction of the circle? A quarter.

You also know:

• A chord connects two points on a circle

New today: an arc's measure is its central angle in degrees — not its length.

Arc Measure Is Not Arc Length

Lengths: and

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

Three Examples: Find the Angle or Arc

Arc : inscribed angle

Inscribed angle : arc

Central angle , same arc: inscribed angle

Identify the intercepted arc first, then apply the theorem.

Inscribed Angles Sharing the Same Arc

Corollary: Inscribed angles intercepting the same arc are congruent.

If , then .

Different arcs: gives , gives .

The Semicircle Theorem: Diameter Makes 90°

An inscribed angle subtending a diameter is always 90°.

Why: a diameter cuts off a arc, so the angle is

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

Scan every circle diagram for a diameter. Mark that angle 90°.

Inscribed Right Triangle: Find the Diameter

→ is a diameter: , radius .

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: One Arc, Two Vertices

Steps: (1) identify arcs; (2) central angle gives arc measure; (3) inscribed angle theorem.

. Arc , so . is on the minor arc, so it intercepts the major arc: , .

SAT Problem 2: Radii Make an Isosceles Triangle

In the same figure, (radii), so is isosceles.

SAT Problem 3: Semicircle and Pythagorean Theorem

Diameter , . Find .

Step 1: on circle + is diameter →

Step 2:

5-12-13 Pythagorean triple

Summary: The Five Key Rules

✓ Arc measure = central angle, in degrees (not length)
✓ Inscribed angle = ½ × intercepted arc (vertex on circle)
✓ Same-arc corollary: same arc, congruent angles
✓ Semicircle theorem: angle on a diameter = 90°; converse holds
✓ Two radii make an isosceles triangle

Watch Out: Five Common Mistakes

Inscribed angle ≠ arc — it equals half the arc
Intercepted arc is opposite the vertex
Vertex inside the circle? Not inscribed
Corollary: only angles on the same arc
Diameter in a diagram → mark the inscribed angle 90°

Coming Up: Tangent Lines and Chord Properties

Next lesson covers:

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

Like today's semicircle problems, many of these come down to finding a right triangle.