What is the measure of the smaller angle?

the hour hand moves 0.5 deg per minute

the minute hand moves 6 deg per minute

at 6:48, the hour hand had moved,

hour hand = 6(60) + 48 = 408 minutes or 0.5(408) = 204 degrees

at 6:48, the minute hand had moved

minute hand = 48 minutes or 6(48) = 228 degrees

the angle formed therefore is

288 deg - 204 deg = 84 degrees

Each hour of the clock face = $\frac{360}{12}$ = 30°

at 6:00 the hour hand is right over the 6 and the minute hand is right over the 12

so the angle between the two hands at 6:00 is 180°

Both hands are constantly moving

so at 6:48 the hour hand has moved thru $\frac{48}{60}$ * 30° = 24°

and the minute hand has gone thru $\frac{48}{60}$ x 360 = 288°

so 288 - 180 = 108° past where the hour hand was at 6:00

so the acute angle formed by the hour and minute hands = 108 - 24 = 84°