Which of the following statements is not true for a rhombus?

Introduction / Context:
This conceptual geometry question asks you to recall the defining properties of a rhombus and identify which given statement does not always hold. Rhombuses are special quadrilaterals with equal sides, and understanding how they differ from other quadrilaterals like rectangles and squares is important for classification and problem solving.

Given Data / Assumptions:

• We are dealing with a rhombus, a quadrilateral with all sides equal.
• Several statements about side lengths, parallelism and diagonals are listed.
• We must determine which statement is not necessarily true for every rhombus.
• Implicitly, standard Euclidean properties of rhombuses apply.

Concept / Approach:
Key properties of a rhombus include:

• All four sides are equal in length.
• Opposite sides are parallel.
• Opposite angles are equal.
• Diagonals bisect each other.
• Diagonals are perpendicular (they intersect at right angles).

However, the diagonals of a general rhombus are not necessarily equal in length. Equal diagonals are a defining property of rectangles and squares. A square is a special rhombus that does have equal diagonals, but not every rhombus is a square.

Step-by-Step Solution:
Step 1: Consider the statement "All sides are congruent". This is true by definition of a rhombus, which has four equal sides. Step 2: Consider "Opposite sides are parallel". A rhombus is a special type of parallelogram, so opposite sides are indeed parallel. Step 3: Consider "Diagonals bisect each other at right angles". In a rhombus, diagonals bisect each other and are perpendicular, so they meet at right angles. Step 4: Opposite angles of a rhombus are equal, just like in any parallelogram, so that statement is also true. Step 5: Consider "Diagonals are equal". This is not a general property of a rhombus. In a rectangle and a square, diagonals are equal, but in a generic rhombus (with unequal angles), diagonals have different lengths. Step 6: Therefore, the statement that is not necessarily true for a rhombus is that its diagonals are equal.

Verification / Alternative check:
You can visualise or draw a fat, slanted rhombus with angles that are not 90 degrees. The diagonals cross at right angles but clearly one diagonal is longer than the other. A square is a special case of a rhombus where all angles are right angles and the diagonals are equal, but since the question is about rhombuses in general, properties specific only to the square cannot be taken as universally true.

Why Other Options Are Wrong:
All sides are congruent: This is the defining property of a rhombus and always holds.
Opposite sides are parallel: A rhombus is a parallelogram, so this is always true.
Diagonals bisect each other at right angles: In a rhombus, diagonals bisect each other and are perpendicular; this is a characteristic feature.
Opposite angles are equal: As a parallelogram, a rhombus has opposite angles equal. This is correct as well.

Common Pitfalls:
Students often confuse the properties of rhombuses, rectangles and squares, assuming that all their properties are interchangeable. It is especially common to mistakenly think equal diagonals apply to all parallelograms or all rhombuses. Remember: equal diagonals are guaranteed in rectangles and squares, but not in general rhombuses; perpendicular diagonals that bisect each other are characteristic of rhombuses and squares, not all parallelograms.

Final Answer:
The statement that is not true for every rhombus is "Diagonals are equal".