According to the Bragg scattering theory, photonic bandgaps (PBGs) in all-dielectric one-dimensional (1-D) photonic crystals (PhCs) are polarization-insensitive. This polarization-insensitive property of PBGs poses a challenge in wide-angle high-performance polarization selection. Herein, we theoretically investigate the angle dependence of the PBGs in a novel kind of 1-D PhC called all-hyperbolic-metamaterial (all-HMM) 1-D PhC entirely composed of HMMs. As the incident angle increases, the PBGs in all-HMM 1-D PhCs exhibit the redshift property under transverse magnetic polarization while exhibiting the blueshift property under transverse electric polarization. Empowered by the polarization-sensitive property of the PBGs, wide-angle high-performance polarization selection can be theoretically achieved. Such a wide-angle polarizer would be useful in liquid crystal displays, quantum interferometers, and Q -switched lasers.