Here you go:
IAH (29°58'50"N 95°20'23"W) SYD (33°34'S 151°06'E) 13828 km
DFW (32°53'49"N 97°02'33"W) SYD (33°34'S 151°06'E) 13796 km
The 777-200ER Range: 14,260km
Although the plane could fly there non-stop, there are prevailing winds, and ETOPS. The routing, Great Circle, would place the 777-200ER flying right through a no-fly zone for 180min ETOPS. Even with 207minutes, it is close. What if there was a delay? They would really be pushing it. Wait for the A340-500/777-200LR to see that type of service.