Proofs with Inscribed Shapes

Explanation

Proofs involving inscribed shapes use known circle theorems to justify angle and arc relationships.

Common facts used in proofs:

• Inscribed angle = ½ intercepted arc

• Opposite angles in cyclic quadrilaterals sum to 180°

• Angles intercepting the same arc are congruent

Proofs may be:

• Two-column proofs

• Paragraph proofs

• Diagram-based reasoning

Quiz (5 Questions)

1. What relationship exists between an inscribed angle and its arc?

2. Why must opposite angles of a cyclic quadrilateral be supplementary?

3. If two inscribed angles intercept the same arc, how are they related?

4. What type of angle intercepts a semicircle?

5. What information is required to prove a quadrilateral is cyclic?

Answer Key

1. Angle equals half the arc

2. Because their arcs sum to 360°

3. They are congruent

4. A right angle

5. Opposite angles are supplementary