Circle Theorems

Vocabulary flashcards defining the essential circle theorems and their geometric properties.

Angle in a semi-circle

The theorem stating that the angle subtended by a diameter at the circumference in a semi-circle is $90^\circ$.

Angle at the circumference theorem

The theorem establishing that the angle subtended at the circumference is half the angle subtended at the centre.

Angles in the same segment

Angles subtended at the circumference from a common chord within the same segment of a circle, which are always equal.

Opposite angles of a cyclic quadrilateral

Pairs of opposing interior angles in a four-sided inscribed polygon that always add up to $180^\circ$.

Radius-tangent theorem

The geometric rule stating that the angle formed between a radius and a tangent at the point of contact is $90^\circ$.

Alternate segment theorem

The theorem stating that the angle between a chord and a tangent is equal to the opposite angle inside the triangle.

Tangents from the same point

The geometric property stating that two tangent segments drawn to a circle from the same external point are equal in length.

Radius through the midpoint of a chord

A radius that intersects the midpoint of a chord and bisects the chord perpendicularly at $90^\circ$.

Angle in a semi-circle

The angle subtended at the circumference by a diameter, which is always $90^\circ$.

Angle at the circumference

The angle subtended at the circumference by an arc, which is half the angle at the centre subtended by the same arc.

Angles in the same segment

Angles subtended at the circumference from a common chord, which are equal to each other.

Cyclic quadrilateral

A four-sided polygon whose vertices all lie on a circle, where the opposite angles always add to $180^\circ$.

Radius and tangent angle

The angle formed between a radius and a tangent at the point of contact, which is always $90^\circ$.

Alternate segment theorem

The theorem stating that the angle between the chord and the tangent is equal to the opposite angle inside the triangle.

Tangents from the same point

Two straight tangents drawn to a circle from the same external point, which are always equal in length.

Radius through the midpoint of a chord

The radius passing through the midpoint of a chord, which bisects the chord at an angle of $90^\circ$.