Measurements of the angle of muons to the shower-core direction and the spread in angle of muons in extensive cosmic-ray showers of energy greater than 10¹⁷ eV are reported. These measurements, along with measurements of the lateral structure function of deep-water detectors and muon density detectors, are compared with rigorous computer simulations. The data are used to derive the average depth of electron cascade maximum as a function of primary particle energy.