Types of quadrilaterals

Quadrilateral: any closed plane figure with exactly four straight sides, four vertices, and four interior angles. Whatever its shape, the four interior angles always add to the same total.

where: $A, B, C, D$ are the four vertices of the quadrilateral, taken in order around it, and the sum is independent of how "regular" or "irregular" the shape looks — a very thin kite and a perfect square obey the same rule. Beyond that shared rule, quadrilaterals split into named families by extra conditions on sides and angles:

• Trapezium: at least one pair of opposite sides parallel. (If the two non-parallel sides are also equal, it is called an isosceles trapezium — a naming detail, not a source of new Class 9 theorems.)
• Kite: two pairs of adjacent sides equal — a sharp contrast with the parallelogram family, where it is opposite sides that match.
• Parallelogram: both pairs of opposite sides parallel. This is the shape this chapter studies in depth, because every extra condition below is layered on top of it.
• Rectangle: a parallelogram in which every angle is $90°$.
• Rhombus: a parallelogram in which all four sides are equal.
• Square: a parallelogram that is simultaneously a rectangle and a rhombus — all sides equal and all angles $90°$.

The clean way to hold this is a chain of specialisation:

with trapezium and kite as separate, single-condition branches that are not generally parallelograms — a trapezium has only one pair of parallel sides, and a kite's equal sides are adjacent, not opposite. Because rectangle, rhombus, and square are each parallelograms with an extra condition bolted on, every property already true of parallelograms — opposite sides equal, opposite angles equal, diagonals bisecting each other — is automatically true of all three without needing separate proof.

This chapter's classification is purely descriptive, not formula-driven — but the one universal fact it leans on, the $360°$ angle sum, does need a reason. Every named quadrilateral above must obey it, and it is worth seeing why rather than accepting it on faith.

Step 1. Take any quadrilateral $ABCD$ and draw diagonal $AC$. This single line splits the quadrilateral into two triangles: $\triangle ABC$ and $\triangle ACD$, sharing the side $AC$.

Step 2. Each triangle's interior angles sum to $180°$, a fact already established before this chapter. So $\angle A_1 + \angle B + \angle C_1 = 180°$ in $\triangle ABC$, and $\angle A_2 + \angle D + \angle C_2 = 180°$ in $\triangle ACD$, where $\angle A_1, \angle A_2$ are the two pieces of $\angle A$ split by the diagonal, and similarly $\angle C_1, \angle C_2$ split $\angle C$.

Step 3. Add the two equations: $(\angle A_1 + \angle A_2) + \angle B + (\angle C_1 + \angle C_2) + \angle D = 180° + 180°$. But $\angle A_1 + \angle A_2$ is exactly the full angle $\angle A$, and $\angle C_1 + \angle C_2$ is the full angle $\angle C$. So the sum collapses to $\angle A + \angle B + \angle C + \angle D = 360°$ — true for any quadrilateral, convex or not, because any diagonal splits it into exactly two triangles.

Step 4. This is why a rectangle's four right angles work out ($90° \times 4 = 360°$), why a rhombus tilted at any angle still totals $360°$, and why, in a square-check like Worked Example below, forcing one angle to $90°$ combined with the parallelogram property forces the rest to follow.

Problem. $ABCD$ is a quadrilateral with all four sides equal, $AB = BC = CD = DA = 5\ \text{cm}$, and $\angle A = 90°$. Diagonal $AC$ is drawn. Name $ABCD$ precisely, and find $AC$.

Step 1. All four sides equal is the defining condition for a rhombus, so $ABCD$ is at least a rhombus. In any parallelogram (and a rhombus is one), opposite angles are equal and consecutive angles are supplementary — so $\angle A = 90°$ forces $\angle B = \angle D = 90°$ too (supplementary to a $90°$ angle), and then $\angle C = 90°$ by the angle-sum check above ($90° \times 3 = 270°$, leaving $90°$ for $\angle C$).

Step 2. All sides equal and all angles $90°$ is precisely the definition of a square. So $ABCD$ is a square with side $5\ \text{cm}$.

Step 3. To find diagonal $AC$: it is the hypotenuse of right triangle $ABC$, with legs $AB = 5\ \text{cm}$, $BC = 5\ \text{cm}$, and the right angle at $B$. By the Pythagorean relation,

Answer: $ABCD$ is a square, and diagonal $AC = 5\sqrt{2}\ \text{cm} \approx 7.07\ \text{cm}$.

Competitive stretch

Step 1. Suppose instead only $\angle A = 90°$ is given, with sides $AB = 5\ \text{cm}$, $BC = 5\ \text{cm}$ but $CD = DA = 5\ \text{cm}$ not assumed equal to $AB$, $BC$ — say $CD = DA = 7\ \text{cm}$ instead, and $ABCD$ is known to be a parallelogram. Now $ABCD$ has opposite sides equal ($AB = CD$, $BC = DA$ would need checking) but not all four sides equal, so it fails the rhombus test; with one right angle forcing all angles to $90°$ (same reasoning as Step 1 above), it is a rectangle, not a square.

Step 2. This is the key discriminator examiners test: a right angle alone gives a rectangle; a right angle and all sides equal together give a square. Always check both conditions before naming the shape.

• Treating parallelogram, rectangle, rhombus, square as four unrelated shapes. They are nested: rectangle and rhombus are each a parallelogram plus one extra condition, and a square has both extra conditions at once — not a separate fourth category.
• Assuming trapeziums or kites inherit parallelogram properties. A trapezium's diagonals do not generally bisect each other, and a kite's opposite sides are not generally equal or parallel — never borrow a parallelogram fact for these two shapes.
• Mixing up "adjacent equal sides" (kite) with "opposite equal sides" (rhombus family). This single distinction is the entire difference between the two shapes — always check which pair of sides is described as equal.
• Forgetting that one right angle in a parallelogram forces all four. Consecutive angles in a parallelogram are supplementary and opposite angles are equal, so a single $90°$ angle cascades to make the whole shape a rectangle (or, with equal sides too, a square) — this shortcut is what makes Worked Example possible without measuring every angle.

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