How come I have the 100, 50,20 distance cycling badges. But not the 5 or 10 mile ones?

Have you done specifically a 5 mile and a 10 mile cycling activity? Not as part of a longer activity.

You have to do a ride that is longer than 5 miles but shorter than 10 miles and another ride that are longer than 10 miles but shorter than 20 miles.

I see, I haven't done any that short. 

But strangely when it recognised my 100 plus rides it then issued the 50 and the 20. 

I thought it was similar to runs where you run a longer distance and it assigns the shorter distance ones automatically. 

I've seen that as well, but I believe it is a mistake that you get the 50 and 20 badges for a 100 mile ride.

For running you should only get one badge for each run and you must run the correct distance. I checked my runs and when I got the 10k badge I didn't get the 5k badge. I got the 5k badge one month later for a 6 km run.

May be that how I got the run badges. As not done any 5k or 10k distances. But done 12,16 and 27 mile ones. So probably got them on separate runs like you say. 

If you like to know you can click the badge and see which activity awarded you with the badge.