1. Shape and Space CIRCLE GEOMETRY

2. Circle Geometry Rule 1 : ANGLE IN A SEMICIRCLE = 90° A triangle drawn from the two ends of a diameter will always make an angle of 90° where it hits the edge of the circle, no matter where it hits. 40° a a = 50°

3. EXERCISES What is the size of each angle? Copy each diagram. 65° a 45° c 63° d 81° b

4. Rule 2 : A TANGENT AND RADIUS MEET AT 90° A tangent is a line that just touches the edge of a curve. If a tangent and radius meet at the same point, then the angle they make is exactly 90°. a = 30° a 60° Circle Geometry

5. EXERCISES What is the size of each angle? Copy each diagram. 47° a 33° b 59° c 71° d

6. Rule 3 : TWO RADII CREATE AN ISOSCELES TRIANGLE These triangles are constructed using two radii. Therefore they are isosceles triangles. a = 30° a 120° Circle Geometry

7. EXERCISES What is the size of each angle? Copy each diagram. 67° a 33° c 110° b 220° d

8. Circle Geometry Rule 4 : THE PERPENDICULAR CORD BISECTOR IS A DIAMETER No matter where you draw the cord, the line that cuts it exactly in half at 90° will go through the center of the circle – the diameter line. a = 36° a 54°

9. EXERCISES What is the size of each angle? Copy each diagram. a 87° b 67° c 41° d 77°

10. Circle Geometry Rule 5 : ANGLES IN THE SAME SEGMENT ARE EQUAL All triangles drawn from a chord will have the same angle where they touch the circle. Also the two angles on opposite sides of the chord add up to 180° a = 50° b = 130° a a b b a + b = 180° 50° a b

11. EXERCISES What is the size of each angle? Copy each diagram. 111° g h 43° a b 31° e f d 76° c

12. Circle Geometry Rule 6 : ANGLE AT THE CENTRE IS TWICE THE ANGLE AT THE EDGE Within the same segment, the angle made at the centre of a circle is exactly double the angle formed by the same chord at the edge of the circle. a = 126° a a a 2a 63° a

13. EXERCISES What is the size of each angle? Copy each diagram. 79° d a 122° 72° b 128° c

14. Circle Geometry ° Rule 7 : OPPOSITE ANGLES OF A CYCLIC QUADRILATERAL ADDS TO 180° A cyclic quadrilateral is a four sided shape with every corner touching the circle. Both pairs of opposite angles add up to 180° a = 93° b = 101° a b d c a + c = 180° b + d = 180° a b 87° 79°

15. EXERCISES What is the size of each angle? Copy each diagram. a b 111° 89° e f 101° 81° c d 72° 61° g h 132° 59°

16. Circle Geometry Rule 8 : EQALITY OF TANGENTS FROM A POINT The two tangents drawn from an outside point always touch the circle at the same distance. This creates a reflective situation with two congruent right angled triangles. a = 41° b = 49° c = 49° 41° a b c

17. EXERCISES What is the size of each angle? Copy each picture. 54° a b c 31° f e d i h 24° g 39° j k l

18. Circle Geometry Rule 9 : ANGLE IN THE OPPOSITE SEGMENT IS EQUAL The angle between the tangent and a chord is equal to the opposite angle in the triangle in the other segment. This is the most difficult to remember! a = 77° b = 71° c = 32° a a b b 71° 77° a b c

19. EXERCISES What is the size of each angle? Copy each picture. 80° 68° a b c 39° 89° g i h 66° 68° a b c 45° f e d 91°