A Square Is Secretly Both a Rectangle and a Rhombus

A square is. It has four right angles, which makes it a rectangle, and four equal sides, which makes it a rhombus — so it satisfies both definitions at once.

Quadrilaterals form a family, with the special ones sitting inside the general ones. This page covers everything in the CBSE Class 8 Mathematics chapter's second part: the parallelogram, the rhombus, kites and trapeziums, and the hierarchy that links them.

A parallelogram is a quadrilateral whose opposite sides are parallel. From that one condition everything else follows:

- Opposite sides are equal
- Opposite angles are equal
- Adjacent angles are supplementary, adding to $180°$
- The diagonals bisect each other

Worked example. One angle of a parallelogram is $70°$. Then:

- The opposite angle is also $70°$
- Each adjacent angle is $180-70=110°$
- So the four angles are $70°$, $110°$, $70°$, $110°$, totalling $360°$

Worked example with a ratio. Two adjacent angles of a parallelogram are in the ratio $2:3$. Since they are supplementary:

$$2x+3x=180\Rightarrow 5x=180\Rightarrow x=36$$

so the angles are $72°$ and $108°$.

Worked example with sides. In parallelogram $ABCD$, $AB=(3x+2)$ cm and $DC=17$ cm. Opposite sides are equal, so $3x+2=17$ and $x=5$.

A sliding gate or a collapsible grille keeps its opposite sides parallel as it moves, which is exactly why the shape it forms stays a parallelogram.

The property a parallelogram does not have is the one to remember. Its diagonals bisect each other but are not equal, and they do not meet at $90°$ — those belong to the rectangle and the rhombus respectively.

A rhombus is a parallelogram with all four sides equal. It inherits every parallelogram property and adds two.

From being a parallelogram:

- Opposite sides parallel, opposite angles equal, adjacent angles supplementary
- Diagonals bisect each other

And in addition:

- All four sides equal
- The diagonals bisect each other at right angles
- Each diagonal bisects the angles of the rhombus

Because the diagonals cross at $90°$ and bisect each other, they cut the rhombus into four right-angled triangles — which is how you find its side from its diagonals.

Worked example. A rhombus has diagonals of $16$ cm and $12$ cm. Half of each is $8$ cm and $6$ cm, so the side is

$$\sqrt{8^2+6^2}=\sqrt{64+36}=\sqrt{100}=10\text{ cm}$$

All four sides are $10$ cm.

Its area comes from the diagonals:

$$\frac{1}{2}\times d_1\times d_2=\frac{1}{2}\times 16\times 12=96{\text{ cm}}^2$$

Worked example with angles. If one angle of a rhombus is $120°$, the opposite angle is $120°$ and the other two are $60°$ each. The diagonal through the $120°$ corner bisects it into two angles of $60°$.

The distinction from a square is exactly one property. A rhombus's diagonals **cross at $90°$ but are not equal**, whereas a square's are both — so the $16$ cm and $12$ cm diagonals above could never belong to a square.

Each is defined by a single distinctive feature.

Kite — two pairs of adjacent (neighbouring) sides are equal, rather than opposite sides.

- So $AB=AD$ and $CB=CD$, with the equal sides next to each other
- One pair of opposite angles is equal — the two between the unequal sides
- The diagonals meet at right angles
- One diagonal (the axis of symmetry) bisects the other and bisects two of the angles
- It has one line of symmetry

Worked example. A kite has sides $AB=AD=5$ cm and $CB=CD=8$ cm, so its perimeter is

$$2(5+8)=26\text{ cm}$$

If its diagonals are $6$ cm and $10$ cm, its area is $\frac{1}{2}\times 6\times 10=30$ cm².

Trapezium — exactly one pair of parallel sides, called the parallel sides, with the other two the non-parallel legs.

- The two angles on the same leg are co-interior between the parallel sides, so they add to $180°$
- An isosceles trapezium has equal legs, and then its base angles are equal and its diagonals are equal

Worked example. In a trapezium with $AB$ parallel to $DC$, angle $A$ is $60°$. Angles $A$ and $D$ lie on the same leg, so

$$\angle D=180-60=120°$$

If angle $B$ is $75°$, then $\angle C=180-75=105°$. Checking: $60+120+75+105=360°$.

A paper kite flown in the sky has exactly this shape, and the cross-section of a canal or an embankment is a trapezium.

The difference that decides which is which is where the equal sides sit. A kite has equal adjacent sides; a parallelogram and rhombus have equal opposite sides — so the same four lengths arranged differently give completely different shapes.

Work from the general to the special, adding one condition at a time.

- Quadrilateral — any four-sided shape, angles totalling $360°$
- Trapezium — exactly one pair of parallel sides
- Parallelogram — both pairs of opposite sides parallel
- Rectangle — a parallelogram with all angles $90°$
- Rhombus — a parallelogram with all sides equal
- Square — a parallelogram with all angles $90°$ and all sides equal, so it is both a rectangle and a rhombus
- Kite — two pairs of equal adjacent sides

What this means in practice:

- Every square is a rectangle, a rhombus and a parallelogram
- Every rectangle and every rhombus is a parallelogram
- A rectangle is not necessarily a square, and a rhombus is not necessarily a square
- A rhombus is a kite too, since its equal adjacent sides satisfy the kite condition

Justifying a classification means naming the properties that hold. A quadrilateral with all sides $7$ cm and all angles $90°$ is a square, because equal sides make it a rhombus and right angles make it a rectangle. One with all sides $7$ cm and angles of $60°$ and $120°$ is a rhombus but not a square, since the angles are not right angles.

And one with diagonals that are **equal and bisect each other at $90°$ must be a square, because equal diagonals give a rectangle and perpendicular diagonals give a rhombus.

The direction of the statements matters, and reversing them is the standard error. "Every square is a rectangle" is true; "every rectangle is a square" is false** — so read carefully which way a question is asking.