= Solution
For a planar lamina in the $xy$-plane and three mutually perpendicular axes through one common point, the <perpendicular axis theorem> is $I_z=I_x+I_y$. Indeed $I_z=\int(x^2+y^2)\,dm$, while $I_x=\int y^2\,dm$ and $I_y=\int x^2\,dm$. Planarity and the common axis intersection are essential.
For two parallel axes, one through the <centre of mass> $G$ and one a perpendicular distance $d$ away, the <parallel axis theorem> is
$$
\boxed{I=I_G+Md^2,}
$$
where $M$ is the total <mass>. Expanding the squared distance in the defining <moment of inertia> integral leaves a cross term proportional to $\int\mathbf r_G\,dm=0$. The second axis need not pass through the body, and uniform <density> is not required, but the reference axis for $I_G$ must pass through its <centre of mass>.
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