Cyclic Quadrilaterals

Lesson 2 of 2: Four Points on a Circle

Learning Objectives

1. Prove that opposite angles of a cyclic quadrilateral sum to .
2. Determine if a given quadrilateral can be inscribed in a circle.

Adding a Fourth Point

• We know any 3 non-collinear points form a unique circle.
• This is why every triangle has a circumcircle.
• What happens if we try to fit a 4-sided shape into a circle?

Misconception: Universal Fit

Watch out: Not all quadrilaterals can be inscribed in a circle!

• 3 points always work.
• 4 points require a special geometric condition.

What is a Cyclic Quadrilateral?

A quadrilateral inscribed in a circle is called a cyclic quadrilateral. All four vertices lie on the circle.

The Inscribed Quadrilateral Theorem

The defining rule for a cyclic quadrilateral:

Opposite angles are supplementary (they sum to ).

Proof: Opposite Angles

• is an inscribed angle looking at arc .
• By the Inscribed Angle Theorem, .

Proof (Continued)

• is an inscribed angle looking at arc .
• .
• Arc + Arc = (the whole circle!)
• .

Example: Finding Unknown Angles

Given: is a cyclic quadrilateral. . Find .

Step 1: Identify opposite angles
and are opposite.

Step 2: Apply the theorem

Misconception: Opposite = Equal?

Watch out: Opposite angles are supplementary, not equal!

• Parallelograms have equal opposite angles.
• Cyclic quadrilaterals have supplementary opposite angles.

Your Turn: Find the Angles

Quadrilateral is inscribed in a circle.
and .

Find the measures of and .

Try this before advancing...

Your Turn (Answer)

Opposite angles sum to .

• is opposite :
  

• is opposite :
  

Your Turn: Is it Cyclic?

A quadrilateral has angles measuring , , , and in order around the shape.

Can this shape be inscribed perfectly in a circle?

Your Turn: Is it Cyclic? (Answer)

No.

• The opposite angles are and (sum = ).
• The other opposite pair is and (sum = ).
• Because they do not sum to , it is not a cyclic quadrilateral.

Synthesis: Triangles vs Quadrilaterals

• Triangles: Any 3 vertices define a single circumcircle.
• Quadrilaterals: A 4th vertex must satisfy the opposite angle rule to join the circle!

Key Takeaways

✓ Cyclic Quadrilaterals are inscribed in a circle.
✓ Their opposite angles sum to .
✓ This is because opposite inscribed angles cut off arcs that combine to make a full circle!
✓ You can use this rule to test if any quadrilateral fits in a circle.