Properties of Quadrilaterals –

Properties of quadrilaterals is a key GCSE Maths topic that appears on Foundation and Higher papers across AQA, Edexcel, and OCR. You need to know the defining features of each quadrilateral — side lengths, angle sizes, diagonal properties, and lines of symmetry — so you can identify shapes, justify geometric proofs, and solve angle problems. This guide covers every quadrilateral you will meet at GCSE with clear summaries, worked examples, and practice questions.

What Is a Quadrilateral?

A quadrilateral is any closed 2D shape with exactly four straight sides and four vertices. The interior angles of every quadrilateral add up to 360°. Beyond that shared property, individual quadrilaterals differ in their side lengths, angle sizes, diagonal behaviour, and symmetry.

Key Properties Summary

Step-by-Step Method

Identifying a Quadrilateral from Its Properties

1. Check whether any sides are parallel — if one pair, consider trapezium; if two pairs, consider parallelogram family.
2. Check whether sides are equal — all four equal narrows to square or rhombus; opposite pairs equal suggests rectangle or parallelogram.
3. Check the angles — four right angles means square or rectangle.
4. Check diagonal properties and symmetry to confirm.

Finding Missing Angles

1. Use the fact that interior angles sum to 360°.
2. Apply any known equal-angle rules (e.g. opposite angles in a parallelogram are equal).
3. Subtract known angles from 360° to find the missing angle.

Worked Example 1 — Foundation Level

Question: A parallelogram has one angle of 65°. Find the other three angles.

Working:

Step 1 — Opposite angles in a parallelogram are equal, so the angle opposite 65° is also 65°.

Step 2 — Co-interior angles between parallel sides add to 180°, so each of the other two angles = 180° − 65° = 115°.

Step 3 — Check: 65° + 115° + 65° + 115° = 360°. Correct.

Answer: The four angles are 65°, 115°, 65°, and 115°.

Worked Example 2 — Higher Level

Question: A kite ABCD has angle A = 110° and angle C = 110°. The two equal angles are at B and D. Find angle B.

Working:

Step 1 — Angle sum = 360°.

Step 2 — A + B + C + D = 360°. Since B = D: 110° + B + 110° + B = 360°.

Step 3 — 220° + 2B = 360°, so 2B = 140°, giving B = 70°.

Answer: Angle B = 70° (and angle D = 70°).

Worked Example 3 — Exam Style

Question: PQRS is a rhombus. Diagonal PR = 10 cm and diagonal QS = 24 cm. Find the perimeter of the rhombus.

Working:

Step 1 — The diagonals of a rhombus bisect each other at right angles. Half of PR = 5 cm; half of QS = 12 cm.

Step 2 — Each side of the rhombus is the hypotenuse of a right-angled triangle with legs 5 cm and 12 cm.

Step 3 — Side = √(5² + 12²) = √(25 + 144) = √169 = 13 cm.

Step 4 — Perimeter = 4 × 13 = 52 cm.

Answer: The perimeter is 52 cm.

Common Mistakes

• Confusing a rhombus with a square. A rhombus has four equal sides but does not require right angles. A square is a special rhombus with 90° angles.
• Forgetting that a square is also a rectangle (and a parallelogram). Properties are inherited — every property of a rectangle also applies to a square.
• Assuming trapeziums have equal diagonals. Only an isosceles trapezium has equal diagonals; a general trapezium does not.

Exam Tips

• Draw and label a quick sketch if the question does not provide a diagram. Mark equal sides with tick marks and right angles with small squares.
• When asked to "explain why" a shape is a specific quadrilateral, state at least two defining properties (e.g. "all sides equal and no right angles, so it is a rhombus").
• Remember that the angle sum of 360° applies to every quadrilateral and is often the starting point for angle calculations.

Practice Questions

Q1 (Foundation): A rectangle has a diagonal of 13 cm and a width of 5 cm. Find the length of the rectangle.

Q2 (Foundation): An isosceles trapezium has angles of 72° and 72° at the base. Find the other two angles.

Q3 (Higher): A rhombus has an angle of 50°. The shorter diagonal is 8 cm. Find the length of the longer diagonal to 1 d.p.

Practise quadrilateral property questions with instant feedback free on GCSEMathsAI.

Related Topics

• Angles in Polygons — interior and exterior angle formulas for any polygon.
• Symmetry: Lines and Rotational — identifying symmetry in shapes.
• Area of 2D Shapes — calculating areas of quadrilaterals.

Summary

Properties of quadrilaterals is tested regularly at both tiers. You must know the side, angle, diagonal, and symmetry properties of squares, rectangles, parallelograms, rhombuses, kites, and trapeziums. Always use the 360° angle sum to find missing angles. In proofs, state specific properties to justify your classification. Understanding how these shapes relate to each other — a square is a special rectangle, which is a special parallelogram — helps you apply the right properties quickly under exam conditions.