Arresting the Enemy King
The GIF is not the below example but the same idea

Arresting the Enemy King

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Let's have a look at this position - White has a Rook on a1 and a King on e1 while Black has a king on e5. It's M15 but you've to know HOW. So let's learn how to mate with a King and Rook.

1. We've to set a jail for the black King with our Rook across the board.

2. We've to move our King up to the Black King.

3. We've to set up another jail with the Rook keeping the King in a corner.

4. If it's possible to make the jail smaller, we should do it

5. If it's impossible to make the jail smaller, we should play a waiting move with the King

6. When the Black king reaches the side, deliver checkmate with the Rook.

Here's an example -

1. Ra4

Just like the King and Queen, the Rook makes a jail for the Black King. He only has 32 squares to move to. Black might want to stay close to the centre so he just keeps moving back and forth and that's when we bring our King.

1. Kd5

2. Ke2

Bringing the King up to the board.

2. Ke5

3. Ke3

Getting the king closer.

3. Kd5

4. Re4

The Rook cuts off half the squares for the King and we need to keep him in jail.

4. Kc5

5. Rd4

4 more squares had been cut off.

5. Kc6

6. Ke4

It's impossible to make the jail smaller so moving the King up.

6. Kc5

7. Ke5

The Kings face each other forcing Black back.

7. Kc6

8. Rd5

Rd5 is better than Rd6+ as the latter allows Kc5. Now Black is restricted to only 9 squares.

8. Kc7

9. Rd6

It's now the final 6 squares for the King.

9. Kb7

10. Kd5

The King comes in again

10. Kc7

11. Kc5

Now Black is forced to the corner.

11. Kb7

12. Rc6 (or Rd7+ but let's go with the former. Now Black has only 4 squares) Ka7

13. Rb6

Now Black has only 2 squares.

13. Ka8

14. Kc6

The King moves in for the final blow

14. Ka7

15. Kc7

Forcing Black to the corner.

15. Ka8

16. Rc6#

In move 16 DO NOT PLAY RB7 AS IT'S STALEMATE AND IT'S A DRAW. But this is how you checkmate with a King and Rook