The image you uploaded shows a geometric problem related to a rhombus. We are asked to prove that the diagonals of a rhombus are perpendicular. The figure is presented in two parts:

• Fig. (a): A triangle $\triangle ABC$, where angle $\theta$ is shown.
• Fig. (b): A rhombus $ABCD$, where the diagonals $AC$ and $BD$ intersect at point $O$.

Step-by-Step Proof:

1. Properties of a Rhombus:

   • In a rhombus, opposite sides are equal and parallel.
   • The diagonals bisect each other at right angles (i.e., the diagonals are perpendicular to each other).
   • Each diagonal splits the rhombus into two congruent triangles.

2. Using Triangles:

   • From Fig. (a), we have $\triangle ABC$ and $\theta$ is the angle between two adjacent sides.
   • In Fig. (b), we see the diagonals intersecting at point $O$. Let’s use this information to show that these diagonals are perpendicular.

3. Congruent Triangles:

   • By symmetry and the properties of a rhombus, we can conclude that the diagonals split the rhombus into four congruent right triangles.
   • Specifically, $\triangle AOB$, $\triangle BOC$, $\triangle COD$, and $\triangle DOA$ are congruent by the Side-Angle-Side (SAS) criterion.

4. Conclusion: Since the diagonals divide the rhombus into right triangles and because of the congruence of these triangles, we can infer that the diagonals intersect at right angles, i.e., they are perpendicular.

Thus, the diagonals of a rhombus are perpendicular.

Would you like further clarification or a more detailed breakdown of any steps? Here are a few related questions that might interest you:

1. What are the properties of a rhombus and how do they relate to other quadrilaterals?
2. How can you prove that the diagonals of a rectangle are equal in length?
3. What is the difference between a rhombus and a square in terms of diagonal properties?
4. How do you prove that the diagonals of a kite are perpendicular?
5. What is the significance of the angle $\theta$ in triangle geometry?

Tip: Remember that proving perpendicularity often involves showing that adjacent triangles are congruent and share common angles.