Focus on the squares on the edges and in the corners of the checker board.
See solution.
Practice makes perfect
To show that the checker board is a square, we need to show that its sides are congruent and that the angles are right angles. To investigate the angles, let's focus on the tiles in the corners of the board.
It is given that the tiles are squares, so all of their angles are right angles. This means that the angles of the board are right angles. Let's now look at the tiles on the edges of the board.
These are congruent square tiles, so their sides are all congruent. The edges of the board all contain eight segments, so the sides of the board are congruent. We can now put together our observations.
The opposite sides of the board are congruent, so according to Theorem 6.9, it is a parallelogram.
All four sides of this parallelogram are congruent, so by definition it is a rhombus.
All four angles of this parallelogram are right angles, so it is a rectangle.
Since the board is both a rectangle and a rhombus, it is a square.
