1. Angles in Circles Objectives: B GradeUse the tangent / chord properties of a circle. A GradeProve the tangent / chord properties of a circle. Use and prove the alternate segment theorem

2. Angles in Circles A line drawn at right angles to the radius at the circumference is called the tangent

3. Angles in Circles O A B P Tangents to a circle from a point P are equal in length: PA = PB OA is perpendicular to PA OB is perpendicular to PB The line PO is the angle bisector of angle APB angle APO = angle BPO The line PO is the perpendicular bisector of the chord AB

4. Angles in Circles Now do these: a 48 o O b O c 36 o 53 o d a = (180-48) ÷ 2 a = 66 o b = 90 o c = 180-(90+36) c = 54 o d = 360-(90+90+53) d = 127 o

5. Angles in Circles The Alternate Segment Theorem The angle between the tangent and the chord is equal to the angle in the alternate segment

6. Angles in Circles Now do these: 58 o e 43 o 86 o g e = 58 o 112 o f The angle at the centre is twice that at the circumference 56 o f = 56 o The angle between the tangent and the chord is equal to the angle in the alternate segment 43 o g = 43+86 g = 129 o

7. Angles in Circles Now do these: 58 o e 43 o 86 o g e = 58 o 112 o f The angle at the centre is twice that at the circumference 56 o f = 56 o The angle between the tangent and the chord is equal to the angle in the alternate segment 43 o g = 43+86 g = 129 o

8. Angles in Circles Worksheet 4 a 48 o O b O c 36 o 53 o d 43 o 86 o g 112 o f 58 o e a = b = c = d = e = f = g =