...but think it's pretty cool that the logo on their current website shows the same orientation of the letters "DAC" that your chip from the 1920's shows:

http://www.dallasathleticclub.org/fw/main/default.asp

Interesting that the other side of the matchbook shows the "A" at the top. Almost as if the D.A.C. Country Club and the Dallas Athletic Club were two separate things. The current addresses for both are the same.