If you're left with only a bishop vs rook, the ending should be drawn.
How should you play to draw this ending?
It is fairly easy to draw the bishop vs rook ending if there aren't any other pieces on the board.
1: Seek shelter with your king in the corner. Go to the corner-square that isn't covered by your bishop.
2: Move your bishop along a diagonal and block possible checks from the rook.
Your defensive setup looks like this:
If the rook tries to check from the side, you interpose the bishop.
The rook can't stay on the first rank, because this would cause a stalemate.
After 2. Rf1 (or anywhere else on the first rank), the bishop can't move.
If the attacker tries to win another way, he will not succeed either.
I'll give you the initial position again.
If White tries 1.Ra2,
You have to move your king, but White can't make any further progress after.
The white king can't play (2.Kb3, Bg8+! winning the rook). So the rook has to play.
After 2.Rh2, (otherwise your king will return to the corner immediately) you have to move the bishop.
This is White's best try. You have to move the bishop and it looks like your king gets chased out of its corner.
After 2...Bg6; 3.Rh1+, Kc2;
White may be able to prevent your king from returning to a1 by playing 4. Ra1.
This isn't a very big problem for you, because your king is now actively placed and you can always run to the other safe corner (h8).
Now let's see what happens if you retreat to the other corner.
You're trapped in the "wrong" corner. This will become clear very soon. After
1. Rc1, the bishop has to interpose with 1...Bf1;
Unfortunately, in this corner the stalemate defence doesn't work.
And although the bishop can't move, the king still has one move left.
It will be mated in the corner after
By now the endgame of bishop vs rook should be an easy draw for you.
You'll run your king to the right corner (the one your bishop can't reach) and you'll make your bishop ready for a stalemate.
This way, you'll always secure a draw in this type of endgame.