In $\mathrm{R}^2$, show the following: (a) The composition of two affine isometries is another affine isometry. (b) The composition of two translations is another translation. (c) The composition of a translation and a rotation (not necessarily centered at the origin) in either order is a rotation. (d) The composition of two plane rotations is either another rotation or a translation. What is the condition for the latter possibility? (e) Every plane translation can be written as the composition of two rotations.