Thales' Theorem and Circumscribed Angles

Lesson 2 of 2: Right Angles and Tangents

Learning Objectives

1. Prove and apply Thales' theorem
2. Explain radius-tangent perpendicularity
3. Understand relationships involving circumscribed angles

A Perfect Right Angle?

• Can an inscribed angle ever be exactly ?
• If an inscribed angle is half of its intercepted arc...
• ...what arc would give us exactly ?

Thales' Theorem

An inscribed angle subtending a diameter is always a right angle.

The Geometry is Dynamic

• Move point anywhere on the circle
• As long as it connects to diameter
• The angle remains

Misconception: Not Just Any Chord!

Watch out: This only works for a diameter.

• If the chord does not pass through the center...
• The angle is not .

The Converse of Thales' Theorem

It works in reverse, too!

• If you have an inscribed angle of ...
• ...the chord it connects to must be a diameter.
• The hypotenuse of the right triangle is the diameter.

Example: Inscribed Right Triangles

Given: A circle with diameter cm. Point is on the circle. cm. Find .

Step 1: By Thales,
Step 2: Use Pythagorean theorem

Quick Check

You are given a circle with points on the edge. You measure and find it is exactly .

Which line segment must be a diameter?

Think for a moment before advancing...

Quick Check (Answer)

The segment connecting the other two points is the diameter.

Therefore, segment must be the diameter.

Tangent Lines

The radius drawn to the point of tangency is perpendicular to the tangent line.

Misconception: Exactly One Point

Watch out: Tangent lines do not "cross" the circle.

• They touch at exactly one point.
• They do not enter the interior of the circle.

Proof by Contradiction

Why must the radius be perpendicular?

• Assume it is not .
• A shorter, perpendicular path to the line must exist.
• That shorter path would be less than the radius.
• That means the line would be inside the circle!

Quick Check

A tangent line touches a circle at point . A radius is drawn to point .

What is the angle between the tangent line and the radius?

Think for a moment before advancing...

Quick Check (Answer)

The angle is (a right angle).

They are perpendicular.

Circumscribed Angles

An angle formed by two tangent lines from an external point.

Circumscribed Derivation

• Quadrilateral has four angles summing to .
• Two angles are (tangent-radius).
• So, remaining.
• Rule: Circumscribed angle + Central angle = .

Example: Finding the Angle

Given: Central angle . Find the circumscribed angle .

Step 1: Apply the supplementary rule

Step 2: Solve

Your Turn

A circumscribed angle measures .

What is the measure of the central angle subtending the same arc?

Try to find the answer before advancing...

Your Turn (Answer)

Step 1: Use the supplementary rule

Step 2: Solve

Key Takeaways

✓ Thales' theorem: Inscribed on a diameter =
✓ Tangent-Radius: Always perpendicular ()
✓ Circumscribed: Supplementary to the central angle

Watch Out for Misconceptions!

Thales' theorem only applies to diameters, not just any chord!
Tangent lines touch exactly once and do not cross the circle.

Next Steps

You've learned how angles work both inside and outside a circle.

Next time, we'll use these tools to explore the inscribed and circumscribed circles of a triangle.