Circle theorems Double Angle Triangles inside Circles Angles connected by a chord Tangents to a circle Cyclic Quadrilaterals

2x2x x This is the ARC o Centre of Circle The Angle x subtended at the centre of a circle by an arc is twice the size of the angle on the circumference subtended by the same arc.

2x x o This is the ARC Centre of Circle Angle subtended at the Centre is twice the angle at the circumference

x x x We are ALL EQUAL This is the Arc Minor Segment Major Segment

o A B CD x 180- x If this angle was 60 0 then angle BCD would be = Points which lie on the circumference of the same circle are called cyclic (or concyclic) points. A cyclic quadrilateral is a quadrilateral with all its four corners (vertices) on the circumference of the same circle.

T A B O TA=TB Tangent

Major Segment Minor Segment ABC E D The Shaded Segment BED is called the alternate segment to the angle CBD The angle between a tangent to a circle and a chord drawn through the point of contact is equal to any angle subtended by the chord at the circumference in the alternate segment

Centre of Circle Diameter

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The angle at the centre

25° x 160° 100° 60° 135° 90° xx xxx Answers 1) 50 2)120 3)180 4)50 5)67.5 6)80 Double angle theorem

Right angles in a semicircle

60° x ° x x x x y y x 100° x 30° 22° y Answers 1) X=30 2)x=18 3)x=45 4)X=40 y=40 5)x=30 y= 120 6)x=22 y=136 x Triangles inside circles

Angles in the same segment

25° x y 15° y z z x y x z x y y z x 25°53° 30° z y x 80° 17° 95° 35° 40° 125° 15° 40° 10° 100° Answers 1) x=25 y=15 2)x=125 y= 40 z=15 3)x=10 y=70 z=100 4)X=105 y=40 z=35 5)x=53 y= 30 z=72 6)x=85 y=80 z=17 Angles connected by a chord (off the same arc)

The tangent and the radius

Two tangents from a point

40° x y z 3 120° x ° x 2 x 35 ° 1 y z Tangents to a circle Answers: 1.x=55 2.x=40 3.x=50 y=50 z=40 4. x=60 y=60 z=30

Angles in a cyclic quadrilateral

x y x y x y 95° 110° 54° 75° 20° 80° x 2a 4b 15° 70° a b 1 25° y z w Answers 1) x=70 y=85 2)x=126 y=105 3)x=100 y=160 4)w=15 x=70 y=65 z= 25 5)a=60 b=36 Cyclic Quadrilaterals

The alternate segment theorem