<p>We study the startup of Poiseuille flow of an incompressible Newtonian fluid in triangular ducts. Exact analytical solutions for the transient velocity field, volumetric flow rate, and wall shear stress distribution are obtained using the eigenfunction superposition method for the hemi-equilateral and right-angled isosceles triangles, two configurations not previously treated in the literature. Together with the equilateral triangle, these are the only triangular geometries that admit trigonometric eigenfunctions of the Laplacian under Dirichlet boundary conditions. The solutions are expressed as double series, extending the classical steady-state results to the time-dependent regime and providing accurate benchmarks for numerical simulations of transient duct flows.</p>