Properties of each quadrilateral — diagonals, angles, symmetry: Question bank

Before the questions, let us fix three words that cause most of the confusion, so no reveal below uses an unexplained term.

Mnemonic The three-switch table (memorise the pattern, not the words)

Shape	bisect?	equal?	perpendicular?
Parallelogram	✔	�’	✗
Rectangle	✔	✔	✗
Rhombus	✔	✗	✔
Square	✔	✔	✔
Isosceles trapezium	✗	✔	✗

A square is the only one with all three switches ON — that is why it feels "most special".

True or false — justify

A trapezium can have two pairs of parallel sides.

False — by definition a trapezium has exactly one pair of parallel sides; two pairs makes it a parallelogram, a different name entirely.

Every parallelogram is a trapezium.

Depends on the definition used — under the strict school definition ("exactly one pair parallel") a parallelogram is not a trapezium; under the inclusive definition ("at least one pair") it is. Always state which convention your syllabus uses.

Every rectangle is a parallelogram.

True — a rectangle has both pairs of opposite sides parallel, which is precisely the definition of a parallelogram; it just adds the extra condition of

90°

angles.

Every square is a rhombus.

True — a square has all four sides equal, which is the definition of a rhombus; a square is simply a rhombus that also happens to have right angles.

Every rhombus is a square.

False — a rhombus needs only equal sides, not right angles; a tilted diamond is a rhombus but not a square.

The diagonals of every parallelogram are equal.

False — they bisect each other but are equal only when the parallelogram is a rectangle (or square); a slanted parallelogram has one long and one short diagonal.

The diagonals of a rectangle are perpendicular.

False — rectangle diagonals are equal but cross at an angle that is

90°

only when the rectangle is a square; for a long thin rectangle the crossing angle is far from

90°

The diagonals of a rhombus are equal.

False — rhombus diagonals are perpendicular and bisect each other, but they are equal only in the square case; a stretched diamond has one long and one short diagonal.

An isosceles trapezium has diagonals that bisect each other.

False — its diagonals are equal in length but they do not bisect each other, because it is not a parallelogram; only parallelograms guarantee mutual bisection.

A quadrilateral with equal diagonals must be a rectangle.

False — an isosceles trapezium also has equal diagonals; "equal diagonals" alone does not force the shape to be a rectangle.

A quadrilateral whose diagonals bisect each other must be a parallelogram.

True — mutual bisection is actually a test for a parallelogram; if each diagonal cuts the other at its midpoint, opposite sides are forced equal and parallel by triangle congruence.

Interior angles of a general quadrilateral always sum to

360°

True for every simple (non-crossing) quadrilateral — one diagonal splits it into two triangles of

180°

each, giving

2 \times 180° = 360°

, regardless of shape.

A parallelogram has a line of symmetry.

False — a general parallelogram has no line of symmetry, only rotational symmetry of order

2

(

180°

turn about its centre). Adding a symmetry line forces it up to a rhombus or rectangle.

A rhombus has exactly two lines of symmetry.

True — its two diagonals are the symmetry lines; a rhombus that is not a square has no other reflection axis (its side-bisectors are not symmetry lines).

Every quadrilateral with four equal sides is a square.

False — four equal sides give a rhombus; it is a square only if the angles are also

90°

Spot the error

A B C D

is a rhombus, so its diagonals are equal — I'll set

A C = B D

Error: rhombus diagonals are perpendicular, not equal. Only the square case makes them equal, so this substitution is illegal for a general rhombus.

P QR S

is a rectangle with

P R ⊥ QS

, therefore area

= \frac{1}{2} P R \cdot QS

Error: rectangle diagonals are not perpendicular. The

\frac{1}{2} d_{1} d_{2}

area formula only works when diagonals cross at

90°

(rhombus, square, or kite), so it cannot be applied to a plain rectangle.

"In trapezium

A B C D

with

A B ∥ C D

, opposite angles are equal, so

∠ A = ∠ C

Error: opposite angles are equal in a parallelogram, not a trapezium. In a trapezium the true relation is that co-interior angles on each leg are supplementary, e.g.

∠ A + ∠ D = 180°

"Diagonals of parallelogram meet at

O

; since

A O = O C

and

B O = O D

, we get

A O = B O

Error: bisection makes each diagonal split into two equal halves, but it does not make halves of different diagonals equal.

A O = B O

would require equal diagonals, i.e. a rectangle.

"An isosceles trapezium is a parallelogram because it has a line of symmetry."

Error: having one line of symmetry does not make a shape a parallelogram. A parallelogram needs both pairs of opposite sides parallel; an isosceles trapezium has only one parallel pair.

"A square has

4

lines of symmetry and rotational symmetry of order

2

Error: the rotational order is wrong — a square has rotational symmetry of order

4

(it looks identical after

90°, 180°, 270°, 360°

turns), not order

2

"Since a rectangle has all angles

90°

, its diagonals bisect the angles."

Error: rectangle diagonals do not bisect the corner angles (unless it is a square); that angle-bisecting property belongs to the rhombus, whose diagonals are also its symmetry axes.

Why questions

Why do the diagonals of a parallelogram bisect each other?

Because triangles

△ A O B

and

△ C O D

are congruent by ASA (alternate angles from the parallel sides plus equal opposite sides

A B = C D

), forcing

A O = O C

and

B O = O D

. See 1.12-Triangle-congruence-criteria.

Why are the diagonals of a rectangle equal?

Because

△ A B C

and

△ B A D

are congruent by SAS (

A B

common,

B C = A D

, and both right angles

90°

), which makes the hypotenuses

A C

and

B D

equal.

Why are the diagonals of a rhombus perpendicular?

Because each diagonal is a line of symmetry, so reflecting across it maps the crossing angle onto its own supplement; an angle equal to its supplement must be

90°

Why does a rhombus's diagonal bisect its corner angles but a rectangle's does not?

In a rhombus the diagonal is a symmetry axis, so it must split the corner angle into two mirror-equal halves; a rectangle's diagonal is not a symmetry axis, so the two halves are generally unequal.

Why does the area formula

\frac{1}{2} d_{1} d_{2}

work for a rhombus?

The perpendicular diagonals cut it into

4

right triangles with legs

\frac{d _{1}}{2}

and

\frac{d _{2}}{2}

; summing their areas gives

4 \cdot \frac{1}{2} \cdot \frac{d _{1}}{2} \cdot \frac{d _{2}}{2} = \frac{1}{2} d_{1} d_{2}

. More area methods in 2.2.04-Area-of-quadrilaterals.

Why does "diagonals bisect each other" prove a shape is a parallelogram, but "diagonals are equal" does not?

Mutual bisection forces opposite-side triangles to be congruent, giving parallel equal opposite sides. Equal length alone says nothing about where they cross, so an isosceles trapezium sneaks in with equal diagonals that don't bisect.

Why does a general parallelogram lack a line of symmetry despite looking balanced?

A reflection would have to map a side onto a parallel side of the same length in the same orientation, but its slant breaks any mirror; only the

180°

rotation (order-

2

rotational symmetry) preserves it. See 1.3.05-Line-symmetry and 1.3.06-Rotational-symmetry.

Why must every simple quadrilateral's angles sum to exactly

360°

no matter how weird its shape?

Because any single diagonal cuts it into two triangles, and each triangle contributes

180°

; the count

2 \times 180° = 360°

never depends on the specific angles. Full argument in 1.2.07-Angle-sum-property-of-quadrilaterals.

Edge cases

Is a shape with all four angles

90°

but sides

3, 4, 3, 4

a square?

No — right angles plus unequal adjacent sides give a rectangle, not a square; a square additionally needs all four sides equal.

Can a "degenerate" quadrilateral have three collinear vertices?

Not as a proper quadrilateral — if three vertices lie on a line, one interior angle becomes

180°

and the figure collapses to a triangle; a valid quadrilateral needs four genuinely non-collinear corners.

What happens to a rectangle's diagonals as it is squished toward a very long thin strip?

They stay equal and keep bisecting each other, but the angle between them shrinks toward

0°

— showing rectangle diagonals are never guaranteed to be perpendicular.

What happens to a rhombus as its corner angle approaches

90°

It smoothly becomes a square — the perpendicular diagonals become equal in length, and the symmetry lines jump from

2

up to

4

at the exact square moment.

Is a "crossed" (self-intersecting) quadrilateral covered by the

360°

angle-sum rule?

No — the

2 \times 180°

derivation assumes a simple (non-crossing) shape; a crossed quadrilateral does not obey the ordinary

360°

sum.

If a quadrilateral has diagonals that are equal, bisect each other, and are perpendicular, what must it be?

A square — those three switches turned on simultaneously uniquely pin down the square, the only shape flipping all three.

Can a trapezium ever have equal diagonals without being isosceles?

No — for a trapezium (one parallel pair), equal diagonals force the legs to be equal, i.e. it becomes isosceles; equal diagonals and the isosceles condition are two sides of the same coin here.

Recall Fast self-test

Which one property, if added to a plain parallelogram, upgrades it to a rectangle? ::: Equal diagonals (equivalently, one right angle). Which one property upgrades a plain parallelogram to a rhombus? ::: Perpendicular diagonals (equivalently, two adjacent sides equal). Which shape has equal diagonals that do not bisect each other? ::: The isosceles trapezium.