Cyclic quadrilaterals: opposite angles
Put the four corners of a quadrilateral ABCD on a circle. Change angles A and B and watch angles C and D, and find the rule for the opposite angles of a cyclic quadrilateral.
Core concept
The opposite angles of a cyclic quadrilateral add up to 180°, because each is half of the arc it stands on and the two arcs make the whole circle; an exterior angle equals the interior opposite angle
Topic: Circle
A quadrilateral whose four corners all lie on a circle is called cyclic. An angle: The amount of turn between two lines that meet, measured in degrees (°). All key words at the circle (an inscribed angle) is half of the central angle on the same arc: the central angle is 2 × the inscribed angle. Angle A stands on the arc BCD and angle C stands on the arc DAB. Together the two arcs make the whole circle, 360°, so A + C = ½ × 360° = 180°. In the same way B + D = 180°. If side BC is carried on past C, the exterior angle at C is 180° − C, which equals A: the exterior angle of a cyclic quadrilateral equals the interior opposite angle.
What you will learn
- State that the opposite angles of a cyclic quadrilateral add up to 180°.
- Explain it using inscribed angles as half of central angles.
- Use the exterior angle of a cyclic quadrilateral.
Before you start
- Know that the angles of a quadrilateral add up to 360°, and what an inscribed angle is.
Materials
- A compass, a ruler, a protractor and paper
Equipment
- None
Procedure
- Record A = 80° and B = 90°.
- Make A = 110° and record.
- Make B = 60° and record.
How this simulation is simplified
- All four corners lie exactly on the circle.