Properties of Quadrilaterals & Their Proofs (Grade 10)

Definitions and full property lists for the parallelogram, rectangle, rhombus, square, trapezium and kite — sides, angles and diagonals — plus how to prove a shape is a parallelogram.

Grade 10 geometry stops asking you to recognise shapes and starts asking you to prove things about them. That means knowing each definition exactly, and knowing which properties you are allowed to quote as a reason.

1The six special quadrilaterals

Matching tick marks show equal sides; dashed lines are the diagonals; the small squares mark right angles.

2Parallelogram

Definition: a quadrilateral with both pairs of opposite sides parallel.

Opposite sides are equal.
Opposite angles are equal.
Co-interior angles are supplementary (add to 180°).
The diagonals bisect each other.

3Rectangle, rhombus and square

Each of these is a parallelogram with something extra — so every parallelogram property is automatically true for them too.

Rectangle = parallelogram + all angles 90°. Its diagonals are equal (and still bisect each other).
Rhombus = parallelogram + all sides equal. Its diagonals are perpendicular and they bisect the angles.
Square = rectangle + rhombus, so it has every property above: equal sides, 90° angles, and diagonals that are equal, perpendicular and bisect the angles.

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Think of it as a family tree: square is both a rectangle and a rhombus, and all three are parallelograms. So proving something about parallelograms proves it for all of them at once.

4Trapezium and kite

Trapezium = one pair of opposite sides parallel. That is all — the diagonals do nothing special.
Kite = two pairs of adjacent sides equal. One diagonal is the axis of symmetry: it bisects the other diagonal perpendicularly and bisects the two angles it passes through.

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A kite's diagonals do not bisect each other — only one diagonal bisects the other. That asymmetry is exactly what exam questions test.

5Proving a quadrilateral is a parallelogram

Any one of these five is enough:

Both pairs of opposite sides are parallel (the definition).
Both pairs of opposite sides are equal.
Both pairs of opposite angles are equal.
The diagonals bisect each other.
One pair of opposite sides is both equal and parallel.

Worked ExampleIn quadrilateral ABCD, AB ∥ DC and AB = DC. Prove ABCD is a parallelogram.

1
Statement: AB ∥ DC (given)
Quote each given fact separately.
2
Statement: AB = DC (given)
3
∴ ABCD is a parallelogram (one pair of opposite sides equal and parallel)
Name the condition you used — the reason carries its own mark.

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Lay every proof out as statement then reason in brackets, one per line, and finish with ∴ and the conclusion. Markers award the reasons separately.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Define a parallelogram.
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A quadrilateral with both pairs of opposite sides parallel.
2
Which quadrilateral has diagonals that are equal AND bisect each other?
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A rectangle (and therefore also a square).
3
Which quadrilateral has perpendicular diagonals that bisect the angles?
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A rhombus (and therefore also a square).
4
How many pairs of parallel sides does a trapezium have?
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One
5
Do a kite's diagonals bisect each other?
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No — only one diagonal bisects the other, and it does so perpendicularly.
6
In parallelogram ABCD, ∠A = 70°. Find ∠C and ∠B.
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∠C = 70° (opposite angles equal); ∠B = 110° (co-interior)
7
State five ways to prove a quadrilateral is a parallelogram.
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Opposite sides parallel; opposite sides equal; opposite angles equal; diagonals bisect each other; one pair opposite sides equal and parallel.
8
Is every rectangle a parallelogram? Is every parallelogram a rectangle?
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Yes to the first; no to the second.