Angle at the centre is double the angle at the circumference HOME Angle at the centre x° Angle at the centre is double the angle at the circumference 2x°

=> Angle at the centre double angle at the circumference PROOF: HOME - First, draw a radius => 2 Isosceles Triangles = (180 – x)/2 x° y° = (180 – y)/2 = 180 – ½x – ½y = 360 – y - x => Angle at the centre double angle at the circumference Q.E.D

Angle in a semi-circle is 90° HOME Angle in a Semi-Circle Angle in a semi-circle is 90°

=> Angle in a semi-circle is 90° PROOF: HOME - First, draw a radius => 2 Isosceles Triangles - Label an angle x° = (180 – x)/2 = 90 - ½x = 180 – x = (180 - (180 – x))/2 = ½x = 90° => Angle in a semi-circle is 90° (or simply apply ‘angle at centre’) Q.E.D

Opposite Angles in a Cyclic Quadrilateral HOME Opposite Angles in a Cyclic Quadrilateral x Opposite angles in a cyclic quadrilateral add up to 180° 2y 2x y

=> Opposite Angles in a Cyclic Quadrilateral add up to 180° PROOF: HOME First, draw in radii x apply ‘angle at centre’ 2y 2x + 2y = 360º 2x 2(x + y) = 360º y x + y = 180º => Opposite Angles in a Cyclic Quadrilateral add up to 180° Q.E.D

Angles created by triangles are equal if they are in the same segment HOME Angles in Same Segment Angles created by triangles are equal if they are in the same segment

PROOF: HOME - First, draw in radii

PROOF: HOME - First, draw in radii apply ‘angle at centre’ x 2x

=> Angles in same segment are equal PROOF: HOME - First, draw in radii y apply ‘angle at centre’ x => 2x = 2y 2y x = y => Angles in same segment are equal Q.E.D

Alternate Segment Theorem HOME Alternate Segment Theorem Angle between a tangent and a chord is equal to the angle at the circumference in the alternate segment Chord Chord Tangent

PROOF: Start with two of the circle theorems HOME Start with two of the circle theorems Angle in a semi circle is 90° Angle between a tangent and the radius is always 90° and now combine them

=> Angles in alternate segments are equal PROOF: HOME For cases when chord isn’t a diameter? Label an angle x 90-x x => Angles in alternate segments are equal Simply apply ‘same segment’ theorem Q.E.D

Tangents to a circle which meet at a point are equal in length HOME Lengths of Tangents A C Tangents to a circle which meet at a point are equal in length (AC = CB) B

=> Tangents equal in length PROOF: HOME - Connect meeting point to centre - Draw radii in apply ‘angle between tangent and radius’ A x2 - r2 r C x O x2 - r2 r B => Tangents equal in length Q.E.D

Perpendicular Tangents HOME Perpendicular Tangents The angle between a tangent and a radius is 90 degrees O