1. Circle Geometry

2. Lesson Objective To discover angle and geometric properties of circles Lesson Success Criteria • Can describe properties of circles • Can solve problems that involve finding angles in circles

3. Circle Revision – parts of a circle minor minor major major

4. 1. Base angles of an isosceles triangle are equal

5. 2. The angle between a tangent and a radius is a right angle

6. 3. The angle at the centre is twice the angle at the circumference

7. 3. The angle at the centre is twice the angle at the circumference Shown are 4 different cases you could come across.

8. 4. Angles from the same arc are equal

9. 4. Angles from the same arc are equal This rule also applies when the centre of the circle is not used. The angles are sharing the same arc – all that matters.

10. 5. The angle in a semi-circle is a right angle This is simply a special case of rule 3: angle at the centre (180°) is twice the angle at the circumference

11. Examples

12. 6. Tangents from a point to a circle are equal • OR = OQ, and OP is common to both • Angles OQP and ORP are right angles • Angles ROP and QOP are equal • Therefore angles OPQ and OPR are equal, and lengths QP and RP must be equal

13. 7. Angle between a chord and a tangent equals the angle in the opposite segment A chord separates a circle into 2 segments.

14. 8. The perpendicular bisector of a chord passes through the circle centre A chord separates a circle into 2 segments.

15. Practice From homework book Page 206 Ex H: Angles in a circle Page 208 Ex I: Further properties of the circle

16. Cyclic Quadrilaterals

17. 1. Opposite angles of a cyclic quadrilateral add to 180° (i.e. are supplementary)

18. 2. The exterior angle of a cyclic quadrilateral equals the opposite interior angle A + B = 180, but B + C = 180 therefore A+B = B+C  A = C

19. Practice From homework book Page 212 Ex J: Cyclic Quadrilaterals