Your opponent is very wise to offer a draw. But you wouldn't be so
wise to accept it.
Yes, it is possible to force a win with a king and a rook versus a
king. Basically what you want to do force your opponent to the edge of
the board. A checkmate will end up looking something like the
.r..K... The capital K is your
........ opponent's king. The
....k... r is your rook and the
........ k is your king. It is
........ your opponent's turn,
........ but he/she is checkmated.
This position (anywhere along the edge of the board) is the only way
you can checkmate your opponent. Your rook must be in the same row or
column as your opponent's king, and your king must be positioned in
such a way that the opponent's king can't move into the adjoining row
There's a good explanation of how you accomplish that here:
Learn to Play
Search on the page for "types of checkmates" or scroll down about
three-fourths of the way. The page explains how to win with just a
king and queen, then explains how you use basically the same procedure
with a rook and king.
You can find another explanation here:
Checkmating with a Lone Rook
If you want to practice and have Java installed in your computer, you
can go here:
Rook and King vs. King
If you play efficiently, you should be able to force checkmate in 15
to 20 moves.
It also may interest you to know you can force a checkmate with two
bishops and a king, or with a bishop and a knight and a king. But you
can't force it with two knights.
Bishop-Knight Simple Checkmate
Bishop Pair Simple Checkmate
Now go win that game!
Mvguy-ga (who was on his high school chess team way back when)
Google search terms used: "rook and king vs. king"
chess "rook and king"