Below is a square that is $3 \times 3$ match sticks. You are given 4 match sticks. Assume all match sticks to be of 1 unit dimension.
The puzzle is to use 4 sticks along with the $3\times 3$ square and divide the outer square into two equal parts that look the same and have same area.
The solution is to lay the blue sticks as below. How to mathematically prove it is correct?
If we rotate either half of either square below by $\space 180˚,\space$ we get a figure of the exact size and shape as the other half. This means they are congruent which is mathematics' way of saying they are equal in all respects. There are infinite other solutions, such as a zig-zag down the middle, but these are sufficient for demonstration.