Diagonals of a Rhombus: Ascertaining whether or not the triangles are congruent

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 $\text{True}$.

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.