Rhombus Diagonal Properties: Investigating Triangle Congruence

To solve this problem, we need to determine if the triangles formed by the diagonals of a rhombus are congruent. A rhombus is a quadrilateral with all sides of equal length, and its diagonals intersect at right angles (90 degrees) and bisect each other. Due to this intersecting property, these diagonals divide the rhombus into four right-angled triangles.

Considering a rhombus with vertices labeled $A$, $B$, $C$, and $D$, where the diagonals intersect at point $O$, such that $AO = OC$ and $BO = OD$. Also, the sides $AB = BC = CD = DA$ due to the definition of a rhombus.

Now, let's focus on the triangles formed: - Triangle $\triangle AOB$ and Triangle $\triangle COD$.

We observe the following:

• The side $AB = CD$ (since all sides of a rhombus are equal).
• Both triangles $\triangle AOB$ and $\triangle COD$ share the perpendicular diagonal $AO = OC$ (since the diagonals bisect each other).
• The angle at the intersection (point $O$ is 90 degrees for both triangles due to the perpendicular nature of the diagonals).

Therefore, by the Side-Angle-Side (SAS) criterion for triangle congruence, these triangles are congruent because they have one side equal, an included right angle, and the other side equal. This analysis applies similarly to any other pair of triangles formed by the division of the rhombus by its diagonals.

Hence, the triangles formed by the diagonals of a rhombus are indeed congruent.

Therefore, the answer is True.