Circle Theorem Proofs l l l SemiCircle Centre

Circle Theorem Proofs l l l Semi-Circle Centre Cyclic Quadrilateral Same Segment Alternate Segment Tangents

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

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

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

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

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

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

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

PROOF: HOME - First, draw in radii

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

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

HOME Ch o d Chor rd Alternate Segment Theorem Angle between a tangent and a chord is equal to the angle at the circumference in the alternate segment Tangent

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

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