Circle Theorems Angles subtended from the same arc. Angles subtended from the same arc are equal.

Example Find the lettered angles for each of the following 50 a (i)

(ii) 18 b

(iii) 22 c 72 d e

(iv) 90 g 65

(v) 105 h i 24

(vi) 84 x y 30

(iii) h 123 105 k i

Angle subtended at the centre The angle subtended at the centre from an arc is double the angle at the circumference.

Find the lettered angles for each of the following Circle Theorems Example Find the lettered angles for each of the following (i) m 84

(ii) n 76

(iii) y 230

(iv) w 68

(v) c 82 a b

(vi) q 24 p r

(ii) 120 x y

(v) 46 t v u

Angles in a semi-circle 90 The largest angle in a semi-circle will always be 90

Example Calculate the size of the lettered angle for each of the following: (i) 28 a b

(ii) a 135 b

(iii) k j 27

(iv) 167 108 m n

Example Find the lettered angles for each of the following: 12 p