Practice Maths

Properties of Quadrilaterals

Key Ideas

Key Terms

parallelogram
A quadrilateral with two pairs of parallel sides; opposite sides equal, opposite angles equal, diagonals bisect each other.
rhombus
A parallelogram with all four sides equal; diagonals bisect each other at right angles.
rectangle
A parallelogram with all four angles equal to 90°; diagonals are equal in length.
square
A rectangle with all four sides equal; a special case of both rectangle and rhombus.
trapezium
A quadrilateral with exactly one pair of parallel sides.
kite
A quadrilateral with two pairs of adjacent equal sides; diagonals are perpendicular.
diagonal
A line segment connecting two non-adjacent vertices of a polygon.
bisect
To divide something into two equal parts.
perpendicular
Meeting at a right angle (90°).

Hierarchy of Quadrilaterals

A square is a special rectangle AND a special rhombus AND a special parallelogram.
A rectangle is a special parallelogram. A rhombus is a special parallelogram.

Hot Tip The interior angles of ANY quadrilateral always add up to 360°. Use this fact to find unknown angles.

Worked Example

Question: ABCD is a parallelogram. ∠A = 112°. Find ∠B, ∠C and ∠D.

Step 1 — Use the property: opposite angles are equal.
∠C = ∠A = 112°

Step 2 — Use the property: co-interior angles are supplementary (add to 180°).
∠B = 180° − 112° = 68°

Step 3 — Opposite angles equal.
∠D = ∠B = 68°

Check: 112 + 68 + 112 + 68 = 360° ✓

What Is a Quadrilateral?

A quadrilateral is any polygon with exactly four sides and four angles. The sum of interior angles in any quadrilateral is always 360°. There are many different types of quadrilaterals, each with its own special properties. Understanding these properties allows you to classify shapes precisely and solve geometry problems more efficiently.

The six main quadrilaterals you need to know are: parallelogram, rectangle, rhombus, square, kite, and trapezium. The first four are related to each other in a hierarchy — a square is a special type of rectangle, which is a special type of parallelogram.

Parallelogram, Rectangle, Rhombus, and Square

A parallelogram has: opposite sides parallel and equal, opposite angles equal, diagonals bisect each other (but are not equal in length and do not meet at right angles).

A rectangle is a parallelogram with all four angles equal to 90°. Its diagonals are equal in length and bisect each other (but don't necessarily meet at right angles).

A rhombus is a parallelogram with all four sides equal in length. Its diagonals bisect each other at right angles (90°) and also bisect the angles of the rhombus.

A square is both a rectangle and a rhombus — all four sides equal AND all four angles 90°. Its diagonals are equal in length, bisect each other at right angles, and bisect all four corner angles (each into 45°).

The hierarchy: Square ⊂ Rectangle ⊂ Parallelogram; and Square ⊂ Rhombus ⊂ Parallelogram.

Kite and Trapezium

A kite has: two pairs of adjacent sides equal (not opposite sides). One pair of opposite angles is equal. The diagonals meet at right angles, and one diagonal bisects the other.

A trapezium has exactly one pair of parallel sides (called the parallel sides or bases). An isosceles trapezium is a special trapezium where the non-parallel sides are equal, the base angles are equal, and the diagonals are equal in length.

Note: a parallelogram has TWO pairs of parallel sides — so a parallelogram is NOT a trapezium (in the Queensland definition, which requires exactly one pair).

Diagonal Properties — A Summary

Diagonals are line segments connecting opposite vertices. Their properties vary by shape:

  • Parallelogram: diagonals bisect each other.
  • Rectangle: diagonals bisect each other AND are equal in length.
  • Rhombus: diagonals bisect each other AND meet at 90°.
  • Square: diagonals bisect each other, are equal in length, AND meet at 90°.
  • Kite: diagonals meet at 90°, one bisects the other.
  • Trapezium: no special diagonal properties (except isosceles trapezium: equal diagonals).

Classifying Quadrilaterals

When asked to classify a quadrilateral, use the most specific name that applies. For example, if a shape has all four sides equal and all four angles 90°, it is a square — not just "a rhombus" or "a rectangle," even though both of those are also technically true. The most specific name is the most informative.

To identify a quadrilateral from given information, check in this order: are all sides equal? Are all angles 90°? Are opposite sides parallel? Are any sides equal? Does it have exactly one pair of parallel sides?

Key tip: A very common exam mistake is saying "a square is not a rectangle" — but a square IS a rectangle (it satisfies all the properties of a rectangle, plus the extra property that all sides are equal). Think of it as a hierarchy: being more specific doesn't exclude you from the more general category. Similarly, every rectangle is a parallelogram, and every rhombus is a parallelogram.

Mastery Practice

  1. Name the quadrilateral described. Fluency

    1. All four sides equal, all angles 90°.
    2. Opposite sides parallel and equal, all angles 90°.
    3. All four sides equal, opposite angles equal, no right angles.
    4. Exactly one pair of parallel sides.
    5. Two pairs of adjacent equal sides, one axis of symmetry.
    6. Opposite sides parallel and equal, opposite angles equal.
    7. Diagonals equal, bisect each other at right angles, all sides equal.
    8. One pair of parallel sides, two pairs of equal angles.

  2. State whether each property belongs to the shape listed. Answer Yes or No and give a reason. Fluency

    1. Rectangle: All sides equal?
    2. Rhombus: All angles 90°?
    3. Parallelogram: Diagonals bisect each other?
    4. Square: Diagonals perpendicular?
    5. Kite: Both pairs of opposite sides parallel?
    6. Trapezium: Exactly one pair of parallel sides?
    7. Rectangle: Diagonals equal?
    8. Rhombus: Diagonals equal?

  3. Find the unknown angles. Give reasons for each step. Fluency

    1. A parallelogram has one angle of 75°. Find the other three angles.
    2. A rhombus has one angle of 40°. Find the other three angles.
    3. A trapezium has three angles: 65°, 115°, 65°. Find the fourth angle.
    4. In a rectangle, a diagonal creates an angle of 38° with the base. Find all four angles at the centre where the diagonals cross.
    5. In a kite, two angles are 110° and 70°. The two equal angles are adjacent to the 70° angles. Find the two unknown angles if the angle sum is 360°.

  4. Use diagonal properties to find unknown lengths. Understanding

    1. In a rectangle, the diagonals have length 13 cm. What is the length of each half-diagonal from the centre?
    2. In a parallelogram, one diagonal is 18 cm and the other is 24 cm. The diagonals bisect each other. Find the distance from each vertex to the centre.
    3. In a rhombus, the diagonals are 10 cm and 24 cm. Use Pythagoras to find the side length of the rhombus.
    4. In a square, the diagonal is 10√2 cm. Find the side length.
    5. In a kite, one diagonal (the axis of symmetry) is 12 cm. The other diagonal is 10 cm and is bisected by the first. Find the length of the two shorter sections of the non-axis diagonal.

  5. ABCD is a quadrilateral. Use the given information to name the most specific type. Explain your reasoning. Understanding

    1. AB ∥ CD, AD ∥ BC, AB = BC = CD = DA, ∠A = 90°.
    2. AB ∥ CD, AD ∥ BC, ∠A = 72°.
    3. AB ∥ CD only, ∠A + ∠D = 180°.
    4. AB = AD, CB = CD, AC ⊥ BD.
    5. All four angles are 90°, AB = CD, BC = AD, AB ≠ BC.

  6. Problem solving with quadrilateral properties. Problem Solving

    1. A park is shaped like a rhombus with diagonals 40 m and 30 m. A path runs along each diagonal. Find the total length of path and the area of the park.
    2. In parallelogram ABCD, ∠A = (3x + 15)° and ∠C = (5x − 11)°. Find x and all four angles of the parallelogram.
    3. A rectangular swimming pool has length (2y + 5) m and width (y + 1) m. The perimeter is 38 m. Find y, the length and the width of the pool.
    4. A designer uses a kite shape for a logo. The kite has two pairs of adjacent sides: 5 cm and 8 cm. The longer diagonal (axis of symmetry) is 10 cm. The shorter diagonal is 6 cm.
      1. Find the area of the kite using Area = (d&sub1; × d&sub2;) ÷ 2.
      2. Verify by splitting the kite into triangles.

  7. Find the unknown angles using algebraic reasoning. Problem Solving

    Algebraic approach. Set up an equation using angle properties of the named quadrilateral, then solve for the unknown.
    1. In a parallelogram, one angle is (4x + 10)° and the opposite angle is (6x − 20)°. Find x and the size of all four angles.
    2. In a rhombus, two adjacent angles are (2y + 15)° and (5y − 12)°. Find y and all four angles of the rhombus.
    3. In a trapezium, the co-interior angles on the same side of a transversal are (3a + 5)° and (2a + 15)°. Find a and both angle sizes. (Hint: co-interior angles between parallel lines add to 180°.)
    4. In a kite ABCD, ∠A = 90°, ∠C = 90°. The two remaining angles B and D satisfy ∠B = 3∠D. Find ∠B and ∠D.

  8. Use quadrilateral properties to find perimeters and side lengths. Problem Solving

    Properties to use. Opposite sides of a parallelogram are equal. All sides of a rhombus or square are equal. Use perimeter = sum of all sides.
    1. A parallelogram has sides (3x + 2) cm and (x + 10) cm on adjacent sides. If the perimeter is 60 cm, find x and the length of each side.
    2. A rhombus has one side of (2m + 5) cm and an adjacent side of (4m − 3) cm. Find m and the perimeter of the rhombus.
    3. A rectangle has a length of (5y − 1) cm and a width of (2y + 3) cm. The perimeter is 54 cm. Find y, the length, and the width.
    4. In a kite, the two longer sides are each (3n + 4) cm and the two shorter sides are each (n + 8) cm. If the perimeter is 72 cm, find n and the two distinct side lengths.

  9. Apply diagonal and geometric properties to solve problems. Problem Solving

    Diagonals as tools. Use the properties of diagonals (bisect, equal, perpendicular) together with Pythagoras’ theorem to find lengths.
    1. A square tile has a diagonal of 20 cm. Find the side length of the tile (leave in surd form or round to 2 decimal places).
    2. A rhombus has diagonals of 16 cm and 12 cm. Find the side length of the rhombus using Pythagoras’ theorem.
    3. In a rectangle, the diagonal is 26 cm and one side is 10 cm. Find the other side using Pythagoras’ theorem. Then find the perimeter and area.
    4. A kite has diagonals of 18 cm (axis of symmetry) and 10 cm. Find the area of the kite. Then find the lengths of all four sides if the axis of symmetry divides the shorter diagonal at the point 6 cm from one end.

  10. Problem solving in real-world and investigative contexts. Problem Solving

    Reasoning. Justify each step using the correct property name (e.g. “opposite angles of a parallelogram are equal”).
    1. A park is shaped like a parallelogram. The longer side is 120 m and the shorter side is 85 m. The park authority wants to fence the entire perimeter. How much fencing is needed?
    2. A logo is designed in the shape of a rhombus with diagonals 24 cm and 10 cm. Find: (i) the area of the rhombus; (ii) the side length; (iii) the perimeter.
    3. Quadrilateral PQRS has angles ∠P = (2x + 5)°, ∠Q = (x + 20)°, ∠R = (3x − 10)°, ∠S = (2x + 15)°. Find x and each angle. Identify the most specific type of quadrilateral PQRS could be, given the calculated angles, and justify your answer.
    4. A tiler is laying square tiles on a rectangular floor. Each tile has a diagonal of 28 cm.
      1. Find the side length of each tile (to 1 decimal place).
      2. The floor is 4.2 m × 3.6 m. How many tiles are needed (to the nearest whole tile)?
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