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

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Circle theorems Double Angle Triangles inside Circles Angles connected by a chord Tangents to a circle Cyclic Quadrilaterals Slide 2 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. Slide 3 2x x o This is the ARC Centre of Circle Angle subtended at the Centre is twice the angle at the circumference Slide 4 x x x We are ALL EQUAL This is the Arc Minor Segment Major Segment Slide 5 o A B CD x 180- x If this angle was 60 0 then angle BCD would be 180 0 -60 0 =120 0 120 0 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. Slide 6 T A B O TA=TB Tangent Slide 7 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 Slide 8 Centre of Circle Diameter Slide 9 This powerpoint was kindly donated to www.worldofteaching.com www.worldofteaching.com http://www.worldofteaching.comhttp://www.worldofteaching.com is home to over a thousand powerpoints submitted by teachers. This is a completely free site and requires no registration. Please visit and I hope it will help in your teaching. Slide 10 The angle at the centre Slide 11 25 x 160 100 60 135 90 xx xxx 1 2 3 6 4 5 Answers 1) 50 2)120 3)180 4)50 5)67.5 6)80 Double angle theorem Slide 12 Right angles in a semicircle Slide 13 60 x 1 23 3 12 72 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 Slide 14 Angles in the same segment Slide 15 25 x 1 2 3 6 4 5 y 15 y z z x y x z x y y z x 2553 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) Slide 16 The tangent and the radius Slide 17 Two tangents from a point Slide 18 40 x y z 3 120 x 4 1 140 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 Slide 19 Angles in a cyclic quadrilateral Slide 20 x y x y x y 95 110 54 75 20 80 x 2a 4b 15 70 a b 1 25 y z w 2 3 45 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 Slide 21 The alternate segment theorem