1) Angles at the centre and circumference
The angle subtended by an arc at the centre is twice the angle subtended at the circumference. More simply, the angle at the centre is double the angle at the circumference.
Calculate the missing angles x and y.
x = 50° x 2 = 100°, and y = 40° x 2 = 80°
On the diagram angle OGK = x and angle OGH = y.
Angle OGK (x) = angle OKG because triangle GOK is isosceles. Lengths OK and OG are both radii.
Angle OGH (y) = angle OHG because triangle GOH is also isosceles. Lengths OH and OG are also both radii.
Angle GOH = 180°-2y (because angles in a triangle add up to 180°).
Angle JOK = 2x (because angles on a straight line add up to 180°).
Angle JOH = 2y (because angles on a straight line add up to 180°).
The angle at the centre KOH (2x + 2y) is double the angle at the circumference KGH (x + y).