In which of the following quadrilaterals are all four sides congruent (that is, equal in length)?

Introduction / Context:
This is a conceptual geometry question about the basic properties of common quadrilaterals. It focuses on which standard quadrilateral has all four sides congruent, meaning all sides have exactly the same length. Recognising these defining properties is essential for quickly classifying quadrilaterals and understanding their symmetry and angle relationships in more advanced problems.

Given Data / Assumptions:

• Options include Rectangle, Parallelogram, Rhombus, Kite and Trapezium.
• We are asked to identify the quadrilateral whose defining property is that all four sides are equal in length.
• We consider standard Euclidean geometry definitions for these shapes.

Concept / Approach:
Each type of quadrilateral has characteristic properties. A rectangle has four right angles and opposite sides equal, but not necessarily all four sides equal. A general parallelogram has opposite sides parallel and equal, but adjacent sides can be different. A rhombus is a special parallelogram with all four sides equal. A kite has two pairs of adjacent equal sides, not all four necessarily equal. A trapezium in general has at least one pair of parallel sides and no requirement on equal side lengths. Therefore, only the rhombus matches the property described in the question.

Step-by-Step Solution:
Consider a rectangle: opposite sides are equal, but adjacent sides are usually different, so not all four sides are equal. Consider a general parallelogram: opposite sides are equal, but there is no requirement that all four sides are the same length. Consider a rhombus: by definition, it is a quadrilateral with all four sides equal and opposite sides parallel. Angles need not be right angles. Consider a kite: it has two pairs of adjacent equal sides. The two pairs can be of different lengths, so not all four sides are equal. Consider a trapezium: it has at least one pair of parallel sides but no requirement of side equality. Thus, among the listed options, only the rhombus always has all four sides congruent.

Verification / Alternative check:
As a useful mental check, remember that a square is a special type of rhombus and also a special type of rectangle. A square has all four sides equal and all angles equal to 90 degrees. If you drop the condition of right angles but keep all sides equal, you get a general rhombus. This confirms that rhombus is the most general quadrilateral type whose key property is four equal side lengths.

Why Other Options Are Wrong:
Rectangle: Requires all angles 90 degrees and opposite sides equal but does not require adjacent sides to be equal unless it is specifically a square. Parallelogram: Only opposite sides are equal and parallel; adjacent sides can differ in length. Kite: Has equal pairs of adjacent sides, not all four sides equal. Trapezium: Defined by having at least one pair of parallel sides, with no condition on side equality.

Common Pitfalls:
Confusing a rectangle with a square and assuming every rectangle has four equal sides. Thinking that all parallelograms must have equal adjacent sides, which is not true in general. Not distinguishing clearly between the definitions of kite, rhombus and square.

Final Answer:
The quadrilateral with all four sides congruent is the rhombus