Let μ and ν be Radon probability measures on ℝ with compact supports. We will consider natural intersections of μ and the image of ν under an isometry f[ratio ]ℝn→ℝn. Our main result gives a lower bound for the packing dimensions of these intersection measures, if μ and ν satisfy some regularity assumptions. Our method, which is originally from [1], is the same as used in [3] where similarities were considered instead of isometries.