State and prove the theorem of perpendicular axes and the theorem of parallel axes.

(N/A) $1$. Theorem of Perpendicular Axes:
This theorem states that the moment of inertia of a planar body (lamina) about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two mutually perpendicular axes lying in the plane of the body and intersecting at the point where the perpendicular axis passes through the body.
Mathematically,$I_z = I_x + I_y$.
Proof: Consider a particle of mass $m$ at point $P(x, y)$ in the $XY$-plane. The moment of inertia about the $X$-axis is $I_x = \sum my^2$,about the $Y$-axis is $I_y = \sum mx^2$,and about the $Z$-axis is $I_z = \sum mr^2$,where $r^2 = x^2 + y^2$. Thus,$I_z = \sum m(x^2 + y^2) = \sum mx^2 + \sum my^2 = I_y + I_x$.
$2$. Theorem of Parallel Axes:
This theorem states that the moment of inertia of a body about any axis is equal to the sum of the moment of inertia of the body about a parallel axis passing through its center of mass and the product of its mass and the square of the distance between the two parallel axes.
Mathematically,$I = I_{cm} + Md^2$,where $I_{cm}$ is the moment of inertia about the center of mass,$M$ is the total mass,and $d$ is the distance between the axes.