A ceramic matrix composite (CMC) component for a combustion turbine engine (10). A blade shroud assembly (30) may be formed to include a CMC member (32) supported from a metal support member (32). The CMC member includes arcuate portions (50, 52) shaped to surround extending portions (46, 48) of the support member to insulate the metal support member from hot combustion gas (16). The use of a low thermal conductivity CMC material allows the metal support member to be in direct contact with the CMC material. The gap (42) between the CMC member and the support member is kept purposefully small to limit the stress developed in the CMC member when it is deflected against the support member by the force of a rubbing blade tip (14). Changes in the gap dimension resulting from differential thermal growth may be regulated by selecting an angle (A) of a tapered slot (76) defined by the arcuate portion.