Properties of Quadrilaterals

A rectangle is defined as a quadrilateral with four right angles. A square has four right angles, so it satisfies this definition. A square also has all four sides equal — that’s an extra property on top of being a rectangle, not a contradiction of it. Just as a Labrador is still a dog, a square is still a rectangle.

Thinking of it another way: if you were checking whether a shape is a rectangle, you would ask “does it have four right angles?” A square passes this test. The fact that it also has equal sides is a bonus, not a disqualification. A square is a special type of rectangle where the sides happen to be equal.

Consider a rhombus — it has four equal sides but its angles are not all 90°. For example, a rhombus with angles 60°, 120°, 60°, 120° has all four sides the same length but is clearly not a square. You can draw one by tilting a square to one side: the sides stay equal, but the right angles are lost.

A square is a special rhombus where all the angles happen to be 90°. Having four equal sides is a necessary condition for being a square, but it is not sufficient — you also need four right angles. Without that second condition, you could have any rhombus.

The angles in any quadrilateral add up to 360°. If three of the angles are 90°, those three sum to 270°. The fourth angle must be 360° − 270° = 90°. So the fourth angle is forced to be a right angle too — you end up with four right angles, not three.

This means it is mathematically impossible to draw a quadrilateral with exactly three right angles. Any attempt to do so will automatically produce a fourth right angle, giving a rectangle (or square). Try to physically draw it: draw three connected line segments with right angles (an open rectangle). Try to close the shape with a straight fourth line. You literally cannot do it without forming a fourth right angle.

Draw a parallelogram where the sides are clearly different lengths — for example, two sides of 5 cm and two sides of 3 cm, with angles of 60° and 120°. This shape has no lines of symmetry at all. If you try to fold it along any line, the two halves do not match up.

Students often assume the diagonals of a parallelogram are lines of symmetry, but they are not — folding along a diagonal does not produce two matching halves. A parallelogram does have rotational symmetry of order 2 (it looks the same after a half turn), but rotational symmetry and line symmetry are different things. Only special parallelograms — rectangles, rhombuses, and squares — have lines of symmetry.

Example: A trapezium with parallel sides of 6 cm and 10 cm, and non-parallel sides of 5 cm and 7 cm.

Another: An isosceles trapezium (the non-parallel sides are equal in length).

Creative: A right trapezium — where one of the non-parallel sides is perpendicular to the parallel sides, giving two right angles. Students rarely think of trapeziums that contain right angles.

Trap: A parallelogram — it has parallel sides, but two pairs of parallel sides, not exactly one. Students may focus on the word “parallel” without checking the “exactly one pair” condition.

Example: A rectangle (e.g., 3 cm × 5 cm — both diagonals are \( \sqrt{34} \) cm long).

Another: A square (e.g., side 4 cm — both diagonals are \( 4\sqrt{2} \) cm long).

Creative: An isosceles trapezium — students rarely realise that the diagonals of an isosceles trapezium are always equal in length (unlike a general trapezium), making this a surprising valid answer.

Trap: A rhombus — students often confuse rhombus diagonal properties. A rhombus has diagonals that are perpendicular and bisect each other, but they are not equal in length (unless it is a square). The “diamond shape” misleads students into thinking everything about the diagonals is special.

Example: A general parallelogram (e.g., sides 4 cm and 7 cm, angles 70° and 110°).

Another: An irregular quadrilateral where all four sides and all four angles are different.

Creative: A non-isosceles trapezium — it has one pair of parallel sides but the non-parallel sides are different lengths, so there is no line of symmetry. Students often overlook trapeziums when thinking about symmetry.

Trap: A kite — it looks “unbalanced” and students may think it has no symmetry, but a kite always has exactly one line of symmetry along its main diagonal (the one connecting the vertices where unequal sides meet).

Example: A rhombus with angles 60°, 120°, 60°, 120°.

Another: A rhombus with angles 80°, 100°, 80°, 100°.

Creative: A very “thin” rhombus with angles 10°, 170°, 10°, 170° — this looks nothing like a square but still has diagonals that bisect each other at right angles.

Trap: A rectangle — its diagonals bisect each other, but they are not perpendicular (they cross at oblique angles). Students may confuse “bisect each other” with “bisect each other at right angles” — these are different conditions.

Example: An arrowhead (or dart). This is a concave quadrilateral.

Another: Any irregular 4-sided shape where one vertex “caves in” towards the center, creating a reflex angle inside the shape.

Creative: An arrowhead where the reflex angle is extremely close to 360° (e.g., 350°), making the shape look almost like a straight line segment with a tiny notch missing.

Trap: A pentagon with a reflex angle. Students often associate “angles bigger than 180°” with complex polygons and automatically draw a 5- or 6-sided shape, forgetting that a simple arrowhead has exactly 4 sides and is therefore a valid quadrilateral.

Every quadrilateral can be split into two triangles by drawing one diagonal. Each triangle has angles summing to 180°, so the total for the quadrilateral is 2 × 180° = 360°. This works for all quadrilaterals — regular, irregular, convex, or concave.

Students who have recently learned the triangle angle sum sometimes apply 180° to quadrilaterals too. The key insight is that a quadrilateral is made of two triangles, doubling the angle sum.

This is true for shapes like rectangles, rhombuses, and general parallelograms, where opposite sides are equal and parallel. However, a kite also has two pairs of equal sides — but the equal sides are adjacent (next to each other), not opposite. A kite is not a parallelogram because it has no parallel sides.

True case: A rectangle with sides 3 cm and 5 cm — two pairs of equal opposite sides, and it is a parallelogram. False case: A kite with sides 4 cm, 4 cm, 7 cm, 7 cm — two pairs of equal adjacent sides, but it is not a parallelogram.

A quadrilateral with four right angles is always a rectangle — but it is only a square if all four sides are also equal. A rectangle where the length and width are different (e.g., 3 cm × 5 cm) has four right angles but is not a square.

True case: A shape with four right angles and all sides 4 cm — this is a square. False case: A shape with four right angles, sides 3 cm and 8 cm — this is a rectangle, not a square.

A trapezium (using the standard UK definition) has exactly one pair of parallel sides. If a quadrilateral had two pairs of parallel sides, it would be a parallelogram, not a trapezium. The single word “exactly” in the definition is what separates these two shape families.

Students sometimes blur the line between trapeziums and parallelograms, thinking both are just “shapes with parallel sides.” The key question is how many pairs of parallel sides: one pair means trapezium, two pairs means parallelogram. These categories do not overlap.

It is true for some shapes, but not all. This highlights a common confusion between rotational and line symmetry.

True case: A square or a rectangle. They both look the same when rotated half a turn (or a quarter turn for a square), and they also have lines of symmetry you can fold along.

False case: A general parallelogram. It has rotational symmetry of order 2 (it maps onto itself after a 180° rotation), but it has exactly 0 lines of symmetry. You cannot fold a slanted parallelogram to make the two halves perfectly overlap.

This is a massive misconception. Students often assume that drawing a diagonal automatically bisects the angles it touches.

True case: A rhombus or a square. Because all four sides are equal, the diagonal forms two congruent isosceles triangles, meaning the corner angles are split perfectly in half.

False case: A rectangle. If you draw the diagonal on a long, thin rectangle (e.g., 3 cm × 10 cm), it is visually obvious that the 90° corner is split into two very unequal angles (e.g., ~17° and ~73°). It absolutely does not cut the 90° angle into two 45° angles!

The student has the “parallelograms must have two different side lengths” misconception. The definition of a parallelogram is a quadrilateral with two pairs of parallel sides — there is no requirement for the sides to be different lengths. A rhombus has two pairs of parallel sides (each pair of opposite sides is parallel), so it is a parallelogram.

The correct answer is square, rhombus, and rectangle. A kite has no parallel sides and a trapezium has only one pair, so neither is a parallelogram. The student was right to exclude those two, but wrong to exclude the rhombus.

The answer is correct — a square does have 4 lines of symmetry — but the reasoning reveals the “one line of symmetry per side” misconception. This reasoning would predict that a rectangle also has 4 lines of symmetry (since it also has 4 sides), which is wrong — a rectangle has only 2.

The 4 lines of symmetry in a square are: two that pass through the midpoints of opposite sides (horizontal and vertical), and two that pass through opposite vertices (the diagonals). They exist because of the specific combination of equal sides and right angles, not because there is “one per side.” A regular pentagon has 5 sides and 5 lines of symmetry, but this “one per side” rule does not work for non-regular shapes.

The student has the “confusing adjacent and opposite equal sides” misconception. In a parallelogram, the equal sides are opposite each other (across the shape) and are parallel. In a kite, the equal sides are adjacent (next to each other) and are not parallel.

A kite is not a parallelogram because it has no parallel sides at all. The student’s reasoning fails because “two pairs of equal sides” can be arranged in two completely different ways: opposite pairs (parallelogram) or adjacent pairs (kite). The arrangement matters, not just the count.

The student has the “opposite angles in all quadrilaterals are equal” misconception. This property only applies to parallelograms, not to all quadrilaterals. The student has taken a special rule and applied it too broadly.

The correct method uses the angle sum: angles in any quadrilateral add up to 360°. So the fourth angle = 360° − 140° − 85° − 60° = 75°. We can verify: 140° + 85° + 60° + 75° = 360° ✓. The student’s answer of 85° gives a total of 140° + 85° + 60° + 85° = 370°, which is impossible.