Why does a light interference pattern suggest that light is transverse?
Well it doesnt per say. Interference patterns are created whenever you diffract a wave, though you can diffract longitudinal as well as transverse waves. Surely polarization would show it was transverse?
How would you use simple mathematics to work out that electromagnetic waves travel at the speed of light?
Distance=Speed*time
Speed=Distance/Time
Simple experiment, use a light of some description, turn it on over a given distance and measure how long it takes to get there
How would you show that light is an electromagnetic wave?
Check it has the qualities that electromagnetic waves do
Travel through a vacuum (at 3*10^8)
Is transverse->can be polarised
Has a magnetic field at 90 degree orientation from the wave