New ${}_5{F}_4$ hypergeometric transformations, three-variable Mahler measures, and formulas for $1\mathrm{/}\pi$

New relations are established between families of three-variable Mahler measures. Those identities are then expressed as transformations for the ${}_5{F}_4$ hypergeometric function. We use these results to obtain two explicit ${}_5{F}_4$ evaluations, and several new formulas for $1\mathrm{/}\pi$.