1. **Problem statement:** We have a circle with center O and points H, J, K, L, M on its circumference. MK is a diameter and is parallel to chord HJ. Given MJ = JL and angle JMK = 38°. 2. **Part (a)(i)(a): Explain why angle HJM = 38°** - Since MK is a diameter, angle JMK = 38° is given. - MJ = JL means triangle MJL is isosceles with MJ = JL. - Because MK is parallel to HJ, angles JMK and HJM are alternate interior angles. - Alternate interior angles formed by a transversal cutting parallel lines are equal. Therefore, $$\angle HJM = \angle JMK = 38^\circ.$$ 3. **Part (a)(i)(b): Explain why angle MJK = 90°** - MK is a diameter of the circle. - By the Thales' theorem, any angle subtended by a diameter on the circumference is a right angle. - Angle MJK is subtended by diameter MK at point J on the circumference. Therefore, $$\angle MJK = 90^\circ.$$