2. Revision- Circle Theorems

3. o A B Theorem 1 The angle at the centre is twice the one at the circumference. C Angle AOB is double angle ACB

4. o A B 84 o xoxo Example Questions 1 Find the unknown angles giving reasons for your answers. o A B yoyo 2 35 o 42 o (Angle at the centre). 70 o (Angle at the centre) angle x = angle y =

5. (180 – 2 x 42) = 96 o (Isos triangle/angle sum triangle). 48 o (Angle at the centre) angle x = angle y = o A B 42 o xoxo Example Questions 3 Find the unknown angles giving reasons for your answers. o A B popo 4 62 o yoyo qoqo 124 o (Angle at the centre) (180 – 124)/2 = 28 0 (Isos triangle/angle sum triangle). angle p = angle q =

6. o Diameter 90 o angle in a semi-circle 20 o angle sum triangle 90 o angle in a semi-circle o a b c 70 o d 30 o e Find the unknown angles below stating a reason. angle a = angle b = angle c = angle d = angle e = 60 o angle sum triangle The angle in a semi-circle is a right angle.Theorem 2 This is just a special case of Theorem 1 and is referred to as a theorem for convenience. Th2

7. Angles subtended by an arc or chord in the same segment are equal. Theorem 3 xoxo xoxo xoxo xoxo xoxo yoyo yoyo Th3

8. 38 o xoxo yoyo 30 o xoxo yoyo 40 o Angles subtended by an arc or chord in the same segment are equal. Theorem 3 Find the unknown angles in each case Angle x = angle y = 38 o Angle x = 30 o Angle y = 40 o

9. The angle between a tangent and a radius is 90 o. (Tan/rad) Theorem 4 o Th4

10. 180 – (90 + 36) = 54 o Tan/rad and angle sum of triangle. 90 o angle in a semi-circle 60 o angle sum triangle angle x = angle y = angle z = T o 36 o xoxo yoyo zozo 30 o A B If OT is a radius and AB is a tangent, find the unknown angles, giving reasons for your answers.

11. Cyclic Quadrilateral Theorem.Theorem 4 The opposite angles of a cyclic quadrilateral are supplementary. (They sum to 180 o ) w y x z Angles x + z = 180 o Angles y + w = 180 o r p s q Angles p + r = 180 o Angles q + s = 180 o Th6

12. 180 – 85 = 95 o (cyclic quad) 180 – 110 = 70 o (cyclic quad) Cyclic Quadrilateral Theorem.Theorem 4 The opposite angles of a cyclic quadrilateral are supplementary. (They sum to 180 o ) 85 o 110 o x y 70 o 135 o p r q Find the missing angles below given reasons in each case. angle x = angle y = angle p = angle q = angle r = 180 – 135 = 45 o (straight line) 180 – 70 = 110 o (cyclic quad) 180 – 45 = 135 o (cyclic quad)