Rhombus Diagonal Properties: Analyzing Triangle Congruence

To determine if the triangles formed by the diagonals of a rhombus are congruent, we proceed with the following analysis:

Step 1: Understanding the properties of a rhombus

• All sides of a rhombus are of equal length.
• The diagonals of a rhombus bisect each other at right angles, meaning each diagonal divides the other into two equal parts.
• The diagonals also form right triangles with each pair of adjacent sides.

Step 2: Applying congruency conditions

Consider the diagonals $AC$ and $BD$ that intersect at a point $E$. The triangles of interest are $\triangle ABE$, $\triangle BCE$, $\triangle CDE$, and $\triangle DAE$.

Each diagonal is bisected by the other, meaning $AE = EC$ and $BE = ED$. Because the diagonals intersect at right angles, each of these triangles is a right triangle.

By the Side-Side-Side (SSS) postulate of congruence:

• $AE = EC$
• $BE = ED$
• The hypotenuse for each set of triangles is a side of the rhombus, which are equal by definition of a rhombus.

Step 3: Conclusion

Thus, all four triangles $\triangle ABE$, $\triangle BCE$, $\triangle CDE$, and $\triangle DAE$ are congruent by SSS postulate, confirming that the triangles formed by the intersection of the diagonals in a rhombus are congruent.

Therefore, the statement that the triangles formed by the diagonals of a rhombus are congruent is $\text{True}$.