1. Circle Theorems  Identify a tangent to a circle  Find angles in circles Tangents Angles in a semicircle Cyclic quadrilateral Major and minor segments Angles at the circumference/centre Alternate segments

2. (1) Tangents  A tangent to a circle is a straight line that touches the circle at one point. TThe point where the tangent touches the circle is called the point of contact. Not tangents Tangents Drawing a radius to any tangent to a circle always produces a 90 0 angle

3. The angles in circles For these triangles : One side is the radius of the circle One side is a tangent to the circle Since these two lines meet at right angles- these are right angle triangles! Right angle

4. Finding angles 32º d d Once you can spot the right angles triangles- you can find missing angles Remember: Angles in a triangle add up to 180 0 90 0 a = 58 0 32 0 + 90 0 + a = 180 0 a 32 + 90 + 58 = 180 d + d + 90 0 = 180 0 45 + 45 + 90 = 180 d = 45 0

5. Angles in a semi circle Use the diameter of a circle as the base of a triangle Any triangle you make using this base will form a right angled triangle

6. (2) Two tangents Two tangents will create Isosceles Triangles You need to identify: Equal pairs of lines (length) Equal pairs of angles

7. Two tangents. 40º Use “Isosceles triangle” to show 70 0 angles 70 0 20 0 Use the right angle (tangent to radius) to calculate the 20 0 Use the isosceles triangle to show 20 0 angle 20 0 Since angles in a triangle add up to 180 0 show the 140 0 140 0

8. (3) Cyclic Quadrilateral A B C D Any 4 points on a circle joined to form a quadrilateral In any cyclic Quadrilateral opposite corners sum to 180 0 So: A + C = 180 0 D + B = 180 0 E F G H So: E + G = 180 0 F + H = 180 0

9. Cyclic Quadrilateral Questions 70º 50º 40º 130 0 What are the missing angles? 110 0 90 0 140 0 90 0

10. (4) Major and minor segments Providing all the angles are in the same segments they will be equal. Major segment Minor segment Any circle can be divided into two unequal parts. These are called segments The larger part is the major segment. The smaller part is the minor segment. The line that divides the circle is called a chord. You can use your chord as the base of a triangle Or… a b c d a = b = c = d

11. (5) Angles at the centre/circumference From a chord you can create a triangle at the centre or at the circumference of the circle. At the centre At the circumference The angle at the centre is twice the size of the angle at the circumference

12. Questions 130º a b 30 0 a = 65 0 b = 60 0 110 0 250 0 C = 125 0 c

13. Putting your rules together 40º Angles at a tangent = 90 0 50 0 An isosceles triangle has two equal angles 40 0 Angles in a triangle add up to 180 0 100 0 Angle at the centre is twice that at the circumference 50 0 Angles round a point add up to 360 0 260 0

14. (6) Alternate segments A A B B Angles in alternate segments are equal