1) The document proves that the angle at the center of a circle is twice the angle at the circumference. 2) It does so by defining various angles within and around a circle with points A, B, O, and P. 3) Through using properties of triangles, angles, and isosceles triangles, it shows that the angle at the center (Angle APB) is equal to half the angle at the circumference (Angle AOB).

1. NJC JH303 Group C By: Pearlyn, Minh, HuiNing, Trisha and Son

2. Prove for property 1: angle at center=2 angle at circumference P O A B

3. Legend: Colours for the different angles Angle APB Angle PBA Angle PAB Angle PBO Angle OBA Angle OAB Angle OAP Angle OPB Angle OBP Angle AOB

4. Prove for property 1: angle at center=2 angle at circumference Angle APB = 180˚­ Angle PBA ­ Angle PAB (Sum of angles in triangle PAB) SinceAngle PBA = Angle PBO + Angle OBA andAngle PAB = Angle OAB − Angle OAP, Angle APB = 180˚− (Angle PBO + Angle OBA) − (Angle OAB − Angle OAP) Therefore, removing the brackets, Angle APB = 180˚− Angle OBA − Angle OAB − Angle PBO + Angle OAP And since OP= OB=OA=radius of circle, Triangle OPB and Triangle OAB are both isosceles triangles. Therefore, in Triangle OPB, Angle OPB andAngle OBP are equal.

5. Prove for property 1: angle at center=2 angle at circumference Hence, Angle PBO = (180˚− Angle POB) ÷ 2 (Base angle of isosceles triangle PBO) Also, Angle OAP = (180˚− Angle AOP) ÷2 (Base angle of isosceles triangle AOP) Moreover, 180˚− Angle OBA − Angle OAB = Angle AOB (Sum of angles in triangle OAB) Therefore, from point 5 of the previous slide,Angle APB = 180˚− Angle OBA − Angle OAB − Angle PBO + Angle OAP, Angle APB = Angle AOB − ((180˚−Angle POB) ÷2) + ((180˚− Angle AOP)÷2) Therefore, removing the brackets, Angle APB = Angle AOB − (-180˚ + 180˚+ Angle POB − Angle AOP) ÷ 2 Summing up, Angle APB = Angle AOB − (Angle POB − Angle AOP) ÷ 2

6. Prove for property 1: angle at center=2 angle at circumference Since Angle AOP − Angle COB = Angle AOB, Point 7 of the last slide can be summarized as Angle APB =Angle AOB − (Angle AOB) ÷ 2. Therefore, Angle APB can be said as ½ Angle AOB which is the angle at the center of the circle. And thus proven angle at center = 2 angle at the circumference because Angle APB = ½ Angle AOB.