In which of the following quadrilaterals are the diagonals always congruent (equal in length)?

Introduction / Context:
This conceptual question asks which type of quadrilateral always has congruent (equal length) diagonals. It tests familiarity with the properties of common quadrilaterals and highlights the special nature of diagonals in each case. Recognising which shapes guarantee equal diagonals is important in many geometry problems.

Given Data / Assumptions:

• We are working with standard definitions of quadrilaterals in Euclidean geometry.
• Options: parallelogram, rhombus, isosceles trapezium, and kite.
• “Diagonals are congruent” means the lengths of the two diagonals are always equal in every quadrilateral of that type.

Concept / Approach:
Recall the general diagonal properties:

• Parallelogram: diagonals bisect each other, but they are equal in length only in special cases such as a rectangle or square, not in every parallelogram.
• Rhombus: all sides are equal and diagonals bisect each other at right angles in special cases, but their lengths are usually unequal unless the rhombus is a square.
• Isosceles trapezium: one pair of opposite sides is parallel, and the non parallel sides (legs) are equal; its diagonals are always equal.
• Kite: two pairs of adjacent equal sides, with diagonals intersecting at right angles in many cases but not necessarily equal in length.

We are looking for the quadrilateral type where equal diagonals are a defining property, not just a special case.

Step-by-Step Solution:
Step 1: Consider a general parallelogram. Its diagonals intersect at their midpoints, but if the adjacent sides are of different lengths, the diagonals are not equal. Only a rectangle or square (special parallelograms) have equal diagonals, so this property is not guaranteed for all parallelograms. Step 2: In a rhombus, all sides are equal, but it typically has one longer and one shorter diagonal; they are perpendicular and bisect each other. Only when the rhombus becomes a square are the diagonals equal, so again, equality is not guaranteed for every rhombus. Step 3: In a kite, the diagonals have a perpendicular intersection and one diagonal often bisects the other, but their lengths are generally different. Equal diagonals are not a defining property of kites. Step 4: In an isosceles trapezium, with one pair of parallel sides and equal non parallel sides (legs), a key property is that the diagonals are always congruent. This follows from the symmetry of the trapezium and can be proven using congruent triangles formed by the diagonal and legs. Step 5: Therefore, among the options, only the isosceles trapezium always has congruent diagonals by definition.

Verification / Alternative check:
A simple sketch of an isosceles trapezium shows that triangles formed by one diagonal and the legs are congruent to the triangles formed by the other diagonal and the legs, due to equal base angles and equal legs. By triangle congruence, the diagonals must be equal in length. In contrast, drawing a typical parallelogram or rhombus that is not a rectangle or square quickly shows visually unequal diagonals.

Why Other Options Are Wrong:
A generic parallelogram can have very different diagonals unless it is a rectangle or square, so equal diagonals are not guaranteed. A rhombus can be tall and thin or nearly square, with diagonals changing accordingly; only a square has equal diagonals. A kite typically has one diagonal longer than the other. Thus none of these quadrilateral types ensure congruent diagonals in every case.

Common Pitfalls:
Some students incorrectly generalise properties of rectangles and squares to all parallelograms and rhombuses, assuming diagonals are always equal. Others mix up properties like “diagonals are perpendicular” (which holds in squares, rhombuses, and many kites) with “diagonals are equal.” To avoid confusion, remember that equal diagonals are characteristic of rectangles, squares, and isosceles trapeziums, but only the isosceles trapezium is explicitly listed among the options here.

Final Answer:
The diagonals are congruent in an isosceles trapezium.