We study the sizes, angular momenta and morphologies of high-redshift galaxies using an update of the Meraxes semi-analytic galaxy evolution model. Our model successfully reproduces a range of observations from redshifts $z=0$-$10$. We find that the effective radius of a galaxy disc scales with UV luminosity as $R_epropto L_{textrm{UV}}^{0.33}$ at $z=5$-$10$, and with stellar mass as $R_epropto M_ast^{0.24}$ at $z=5$ but with a slope that increases at higher redshifts. Our model predicts that the median galaxy size scales with redshift as $R_e propto (1+z)^{-m}$, where $m=1.98pm0.07$ for galaxies with $(0.3$-$1)L^ast_{z=3}$ and $m=2.15pm0.05$ for galaxies with $(0.12$-$0.3)L^ast_{z=3}$. We find that the ratio between stellar and halo specific angular momentum is typically less than one and decreases with halo and stellar mass. This relation shows no redshift dependence, while the relation between specific angular momentum and stellar mass decreases by $sim0.5$ dex from $z=7$ to $z=2$. Our model reproduces the distribution of local galaxy morphologies, with bulges formed predominantly through galaxy mergers for low-mass galaxies, disc-instabilities for galaxies with $M_astsimeq10^{10}$-$10^{11.5}M_odot$, and major mergers for the most massive galaxies. At high redshifts, we find galaxy morphologies that are predominantly bulge-dominated.
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