Lattice-matched graphene on hexagonal boron nitride is expected to lead to the formation of a band-gap but requires the formation of highly strained material and has not hitherto been realised. We demonstrate that aligned, lattice-matched graphene can be grown by molecular beam epitaxy using substrate temperatures in the range 1600-1710 °C and coexists with a topologically-modified moiré pattern, and with regions of strained graphene which have giant moiré periods up to ~80 nm. Raman spectra reveal narrow red-shifted peaks due to uniform isotropic strain, while the giant moiré patterns result in complex splitting of Raman peaks due to strain variations across the moiré unit cell. The latticematched graphene has a lower conductance than both the Frenkel-Kontorova-type dislocation lines, and also the topological defects where they terminate. We relate these results to theoretical models of band-gap formation in graphene/boron nitride heterostructures.