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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 $R_ξ$ 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.