The diagonals of rhombus are equal ( True / False )

Answer

To determine whether the statement "The diagonals of a rhombus are equal" is true or false, we will analyze the properties of a rhombus step by step.

### Step-by-Step Solution:

1. **Understanding the Definition of a Rhombus**:
A rhombus is a type of quadrilateral where all four sides are of equal length.

**Hint**: Remember that in a rhombus, the sides are equal, but this does not imply anything about the diagonals.

2. **Drawing a Rhombus**:
Draw a rhombus and label its vertices as A, B, C, and D. Draw the diagonals AC and BD.

**Hint**: Visualizing the shape can help you understand the relationships between the sides and angles.

3. **Identifying the Diagonals**:
The diagonals of the rhombus are AC and BD. We need to investigate whether these diagonals are equal.

**Hint**: Focus on the properties of the triangles formed by the diagonals.

4. **Analyzing the Triangles Formed by the Diagonals**:
The diagonals of the rhombus intersect at point O, dividing the rhombus into four triangles: AOB, BOC, COD, and DOA.

5. **Using Triangle Congruence**:
Consider triangles AOD and BOC. We know:
- AD = BC (sides of the rhombus are equal)
- CD = CD (common side)
- We need to check the angles.

**Hint**: Use the properties of congruence criteria to analyze the triangles.

6. **Checking the Angles**:
The angles ∠AOD and ∠BOC are not necessarily equal in a rhombus. They can be different unless the rhombus is a square.

**Hint**: Remember that in a square, all angles are right angles, which is not the case for a general rhombus.

7. **Conclusion**:
Since the angles in triangles AOD and BOC are not equal, we cannot use the Side-Angle-Side (SAS) criterion for triangle congruence. Therefore, we conclude that the diagonals AC and BD are not equal in a rhombus.

**Final Statement**: The statement "The diagonals of a rhombus are equal" is **False**.