  We analyze the structure of scattering amplitudes of the Kaluza-Klein (KK)
gravitons and of the KK gravitational Goldstone bosons in the compactified 5d
General Relativity (GR). Using a general $R_{\xi}$ gauge-fixing, we study the
geometric Higgs mechanism for the massive spin-2 KK gravitons. We newly propose
and prove a Gravitational Equivalence Theorem (GRET) to connect the scattering
amplitudes of longitudinal KK gravitons to that of the KK gravitational
Goldstone bosons, which formulates the geometric gravitational Higgs mechanism
at the scattering $S$-matrix level. We demonstrate that the GRET provides a
general energy-cancellation mechanism guaranteeing the $N$-point longitudinal
KK graviton scattering amplitudes to have their leading energy dependence
cancelled down by a large power factor of $E^{2N}$ ($N \geq 4$) up to any loop
order. We propose an extended double-copy approach to construct the massive KK
graviton (Goldstone) amplitudes from the KK gauge boson (Goldstone) amplitudes.
With these we establish a new correspondence between the two types of energy
cancellations in the four-point longitudinal KK amplitudes at tree level:
$E^4\to E^0$ in the KK gauge theory and $E^{10} \to E^2$ in the KK GR theory.
