Rigidities rhymes with:
annuities, authorities', authority's, availabilities, banalities, causalities, cavities, celebrities, celebrity's, charities', christianity's, community's, complexities, complicities, eccentricities, entities, entities', facility's, gratuities, humanities, humanity's, identities, illegalities, immunities, indemnities, inequalities, infidelities, infirmities, insecurities, instabilities, legalities, localities, majorities, maturities, municipalities, nationalities, necessities, niceties, obscenities, personalities, principalities, rarities, sensibilities, sensitivities, trinity's, universities', varity's, vulnerabilities
Rigidities sounds like:
racists, rackets, racquets, ragtag, rastus, reacts, recedes, recites, regattas, rejects, requests, requisites, resets, resides, residues, resists, restage, restates, restock, rests, richts, rickets, ricketts, right's, righteous, rightish, rights, rights', riksdag, roasts, rocket's, rockets, rockettes, roget's, rosettes, rosewood's, rostuca, rushdie's, russets, rust's, rustic, rusts, rusty's, rustys