Circle Theorems and Proofs
Circle theorems describe relationships between angles, chords, and arcs in circles. These properties can be proven using congruence and properties of isosceles triangles.
What You Need to Know
Key Concept Diagram
Angle at centre = 2 x angle at circumference subtended by same arc
Angles in the same segment (same arc) are equal
Angle in a semicircle = 90 degrees (diameter subtends a right angle)
Opposite angles of a cyclic quadrilateral are supplementary (sum to 180)
Key Vocabulary
Chord
A line segment joining two points on a circle
Arc
A part of the circumference of a circle
Cyclic quadrilateral
A quadrilateral with all four vertices on a circle
Subtend
To be opposite to and delimit an angle or arc
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
The angle at the centre of a circle is 80 degrees. What is the angle at the circumference subtending the same arc?
Question 2
An angle in a semicircle is always:
Question 3
In a cyclic quadrilateral, opposite angles sum to:
Key Concepts Summary
- ●Angle at centre = 2 x angle at circumference subtended by same arc
- ●Angles in the same segment (same arc) are equal
- ●Angle in a semicircle = 90 degrees (diameter subtends a right angle)
- ●Opposite angles of a cyclic quadrilateral are supplementary (sum to 180)