Introduction:
In this question, we need to prove that the four triangles formed by the diagonals and sides of a rhombus are congruent. A rhombus is a quadrilateral with all sides of equal length. The diagonals of a rhombus bisect each other at right angles.

Proof:
To prove that the four triangles are congruent, we will use the concept of side-angle-side (SAS) congruence.

Step 1: Diagonals of a rhombus bisect each other at right angles:
The diagonals of a rhombus bisect each other at right angles. This property is given in the question.

Step 2: Side 1:
Let's consider one of the triangles formed by the diagonals and sides of the rhombus. We'll label the vertices of the triangle as A, B, and C.

- Side 1: AC (one of the sides of the rhombus)

Step 3: Side 2:
- Side 2: AB (diagonal of the rhombus)

Step 4: Angle:
- Angle: ∠BAC (formed by side AC and diagonal AB)

Step 5: Side 3:
To prove that the triangles are congruent, we need to show that the corresponding sides and angles of the triangles are congruent.
- Side 3: BC (another side of the rhombus)

Step 6: Side 4:
- Side 4: CD (diagonal of the rhombus)

Step 7: Angle:
- Angle: ∠BCD (formed by side BC and diagonal CD)

Step 8: Proving congruence:
Now, let's compare the corresponding sides and angles of the triangles.

- Side 1: AC is congruent to side 3: BC (rhombus property)
- Side 2: AB is congruent to side 4: CD (rhombus property)
- Angle ∠BAC is congruent to angle ∠BCD (vertical angles formed by diagonals)
- Angle ∠ABC is congruent to angle ∠CBD (rhombus property)

Conclusion:
By using the SAS (side-angle-side) congruence criterion, we have proven that the four triangles formed by the diagonals and sides of the rhombus are congruent.