Question

given ef intersects gh at point D and ef is perpendicular to gh, what is the measure of hdf

Answer 1

Below is an image of what you have described. Since EF intersects GH perpendicularly - all resulting angles will be 90° by the definition of intersection of perpendicular lines.

This is to say that, Angle GDE, GDF, HDF, and HDE are all 90° angles.