Can a rook and a king defeat a bishop and a king?
Yes, but only if the stronger side (the player with the rook) can manage to drive the king into the corner that has the same color as the bishop.
If you have the bishop then make sure you go with your king into a corner of the board that has the OPPOSITE COLOR of your bishop.
This means if your bishop is running on black squares you go with your king in a corner that has a white square.
See diagram below, this position is a draw as the black king is in a corner that has a white square as his bishop runs on black squares.
Draw – Black king has reached the correct corner
If White has to move now he must leave the last row with the rook or it will be a stalemate, as Black can’t make a legal move.
In these kind of positions Black just moves with his bishop as soon as the rook leaves the last rank and the black king stays in the corner.
How White wins
If the black king is badly placed near the wrong corner White has the chance to win as he drives the king into the corner that has the same color as the bishop.
The black king is cut off by the rook.
1. Kf7 (White King moves closer opposite the black king.) 1…Bf4
2.Rh3 + Bh6
3. Rh4 (White makes a waiting move!) Black is in Zugzwang!
Black is checkmate