Central and Inscribed Angles

Lesson 1 of 2: Arcs and Angles on the Circle

Learning Objectives

1. Define central angles and inscribed angles
2. State and apply the Inscribed Angle Theorem
3. Understand the relationship between intercepted arcs and these angles

How Do We Measure Parts of a Circle?

• We know a full circle is
• What happens when we take a "slice" of a circle?
• How do we measure the curved edge of that slice?

Circle Vocabulary Review

• Radius: Center to edge
• Chord: Edge to edge
• Diameter: Chord through the center
• Arc: A continuous curve on the edge

Central Angles and Arc Measure

The arc measure equals the central angle that intercepts it.

Arc Measure vs. Arc Length

• Arc Measure: A rotation angle (degrees)
• Arc Length: A physical distance (cm, inches)
• Two circles can have the same arc measure but different arc lengths!

Example: Finding Arc Measure

Given: A circle with center . .

Step 1: Identify the central angle

Step 2: Apply the definition
The minor arc equals the central angle.

Quick Check

If the minor arc measures , what is the measure of the major arc ?

Think for a moment before advancing...

Quick Check (Answer)

Step 1: A full circle is

Step 2: Subtract the minor arc

The major arc measures .

Equal Chords and Equal Arcs

• In the same circle, congruent chords intercept congruent arcs.
• If chord chord , then arc arc .

Discovery: Measuring from the Edge

What happens if we move the vertex from the center to the edge?

Discovery Results

No matter where point is, the angle is always half the central angle.

The Inscribed Angle Theorem

An inscribed angle is half the measure of its intercepted arc.

Misconception: Equal vs. Half

Watch out: Do not confuse central and inscribed angles!

• Central Angle: Equals the arc
• Inscribed Angle: Half the arc (needs to stretch further)

Proof of Case 1 (Diameter)

• Assume one side of the angle is a diameter.
• is isosceles ().
• Base angles are equal: .

Proof of Case 1 (continued)

From the previous slide: has base angles

• Exterior angle .
• Inscribed angle is exactly half the central angle .

Cases 2 and 3

• Case 2: Center is inside the angle.
• Case 3: Center is outside the angle.
• Both proved by drawing a diameter and using Case 1!

Corollary: Angles on the Same Arc

All inscribed angles subtending the same arc are equal.

Misconception: Same Chord vs. Same Arc

Watch out: Angles on the major vs minor arc are different!

• Same arc = equal angles
• Opposite arcs = supplementary angles ( sum)

Example: Finding Unknown Angles

Given: Intercepted arc is . Find the inscribed angle .

Step 1: Identify angle type
It is an inscribed angle.

Step 2: Apply theorem

Your Turn

An inscribed angle measures .

1. What is the measure of the intercepted arc?
2. What is the central angle subtending the same arc?

Try to find both answers before advancing...

Your Turn (Answers)

1. Intercepted Arc:

2. Central Angle:

Key Takeaways

✓ Central angles equal the intercepted arc
✓ Inscribed angles equal half the intercepted arc
✓ Inscribed angles on the same arc are equal

Watch out: Don't assume angles on opposite sides of a chord are equal!

Next Steps

Today we learned about angles inside the circle.

Next lesson, we'll discover a special right angle theorem and explore angles that exist outside the circle!