Chapter 9, Problem 4CUR

To determine

To prove: the diagonals of a rhombus divide the rhombus into four congruent triangles.

Explanation of Solution

Consider the following diagram:

In the above diagram a rhombus $ABCD$ is shown and $M$ is the point at which its diagonals intersects. As it is already known that the diagonal of the rhombus are bisectors of each other, $AM=MC$ and $BM=MD$ and also the lengths of sides of the rhombus are also equal, and every set of 2 triangles share one common bisector.

Therefore in $\Delta AMB$ and $\Delta CMD$

  $\begin{array}{l}AM=MC\\ BM=MD\end{array}$

And $AB=DC$

So, $\Delta AMB$ and $\Delta CMD$ are congruent by SSS postulate. Similar proof can be given for all other triangles.

Therefore all four triangles are congruent by SSS postulate.