Most tested M5.3

Properties of Triangles and Quadrilaterals

This topic covers the essential properties of common triangles and quadrilaterals. Success in ESAT questions depends on rapidly identifying a shape from given information and applying its specific rules about side lengths, angles, and diagonals to find missing values.

Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.

Key points

  • Shapes can be classified hierarchically. A square is a special type of rectangle and rhombus. A rhombus and a rectangle are both special types of parallelogram.
  • Diagonals are powerful tools. They intersect at right angles in a kite, rhombus, and square. They bisect each other in any parallelogram (including rectangles, rhombuses, squares).
  • Symmetry reveals properties. An isosceles triangle has one line of symmetry, which bisects the top angle and the base. A rhombus has two lines of symmetry along its diagonals.
  • The sum of interior angles in any quadrilateral is 360°. Adjacent angles in a parallelogram are supplementary (add to 180°).
  • In an isosceles triangle, the two angles opposite the equal sides are themselves equal. In an equilateral triangle, all sides are equal and all angles are 60°.

Diagram

Triangle diagramTriangle ABC, sides AB=x, CA=x, BC=base, angles B=θ, C=θ. xbasexABθCθ
An isosceles triangle has two equal sides (marked x) and two equal base angles (θ). The three interior angles always sum to 180°.

Formulae

Area = 1/2 × (a + b) × h

To find the area of a trapezium, where 'a' and 'b' are the lengths of the parallel sides and 'h' is the perpendicular height between them.

Area = 1/2 × p × q

To find the area of a kite or a rhombus, where 'p' and 'q' are the lengths of the two diagonals. This is often faster than using base times height.

Definitions

Parallelogram
A quadrilateral with two pairs of parallel sides. Key consequences: opposite sides are equal, and opposite angles are equal.
Trapezium
A quadrilateral with exactly one pair of parallel sides.
Kite
A quadrilateral with two distinct pairs of equal-length adjacent sides. Key consequences: one pair of opposite angles are equal, and diagonals are perpendicular.
Rhombus
A parallelogram where all four sides are equal length. Key consequences: diagonals bisect each other at 90° and also bisect the corner angles.

Worked example

A quadrilateral PQRS is formed by joining two identical isosceles triangles, PQR and PSR, along their common base PR. If angle PQR is 100°, what is the name of the quadrilateral and the size of angle QPS?

  1. 1

    Identify the properties of the triangles.

    PQR is isosceles, so PQ = QR.

    PSR is identical, so PS = SR.

    The common base is PR.

  2. 2

    Combine the properties for the quadrilateral.

    We have PQ = QR and PS = SR

    This fits the definition of a kite, which has two pairs of equal-length adjacent sides.

  3. 3

    Analyse the angles.

    In triangle PQR, the base angles are QPR and QRP.

    Since the angles in a triangle sum to 180°, angle QPR + angle QRP + 100° = 180°.

    So, 2 × angle QPR = 80°, which means angle QPR = 40°.

  4. 4

    Since triangle PSR is identical to PQR, angle SPR is also 40°.

  5. 5

    The total angle QPS is the sum of angles QPR and SPR.

  6. 6

    Calculate the final angle:

    Angle QPS = 40° + 40° = 80°

Answer: The quadrilateral is a kite, and angle QPS is 80°.

Common mistakes

  • ×Forgetting a shape's full set of properties. For example, knowing a rhombus has equal sides but forgetting that its diagonals bisect the angles, which is often the key to solving a problem.
  • ×Incorrectly applying properties to more general shapes. A common mistake is assuming a general parallelogram has perpendicular diagonals; this is only true for a rhombus or a square.
  • ×Confusing a kite with a parallelogram. A kite has equal *adjacent* sides, whereas a parallelogram has equal *opposite* sides.

No-calculator tips

  • ✓Always draw a quick sketch. Visualising the geometry helps you see which angles should be acute or obtuse and which lines are longer, preventing simple calculation errors.
  • ✓Deconstruct shapes into right-angled triangles. Diagonals of kites, rhombuses, and rectangles often form right-angled triangles, allowing you to use basic geometric rules without needing a calculator.
  • ✓Use the 360° sum for quadrilaterals. If you know three angles, you can always find the fourth. For parallelograms, just remember any two adjacent angles sum to 180°.

Read this topic in the official UAT-UK ESAT guide →

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