Star Alliance member Ethiopian Airlines has ticket prices to Cape Town and Johannesburg, South Africa from $767 to $786 round trip for travel September to April 2018.
Ethiopian flies LAX to Dublin, Ireland nonstop. A short connection time in Dublin, then fly Ethiopian to Addis Ababa, Ethiopia ADD. From ADD, connect for Ethiopian flights to Johannesburg JNB or Cape Town CPT.
Ethiopian Airlines $786.08 round trip
Los Angeles LAX – Cape Town, South Africa CPT
Ethiopian Airlines $781.58 round trip
Los Angeles LAX – Johannesburg, South Africa JNB
I found slightly lower Momondo.com ticket prices at $767 round trip.