Chapter 8 · NCERT Class 9 Mathematics

Quadrilaterals

Four sides, four angles, one rule: they always add up to 360°. Add the right extra property — parallel sides, equal sides, right angles — and a plain quadrilateral becomes a parallelogram, rectangle, rhombus or square. Tap each shape to see what makes it special.

📐 3 topics⏱ ~25 min📝 12-question quiz

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The quadrilateral family

Each special name adds exactly one more property on top of the last. Tap each shape to see what defines it and how its diagonals behave.

Explore · The family of four-sided figurestap a shape

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The three big ideas

A quadrilateral is a closed figure with 4 sides, 4 vertices and 4 angles.
Angle-sum property: a diagonal splits it into two triangles, so its four angles always add up to 360° (2 × 180°).
The family by type: trapezium (one pair of parallel sides) → parallelogram (both pairs parallel) → rectangle, rhombus → square (the most special).

Worked example. Three angles of a quadrilateral are 90°, 80° and 100°. Find the fourth angle.

The four angles add to 360°, so the fourth = 360° − (90° + 80° + 100°) = 360° − 270° = 90°.

In a parallelogram, both pairs of opposite sides are equal and parallel, the opposite angles are equal, and the diagonals bisect each other.
Rectangle — a parallelogram with all four angles 90°; its diagonals are equal.
Rhombus — a parallelogram with all four sides equal; its diagonals bisect each other at right angles.
Square — all sides equal AND all angles 90°: it is both a rectangle and a rhombus.

Common mistake: Every square is a rectangle AND a rhombus, because it has every property they require. But the reverse is false — not every rectangle or rhombus is a square (a long rectangle lacks equal sides; a tilted rhombus lacks right angles).

Mid-point theorem: the segment joining the mid-points of two sides of a triangle is parallel to the third side and half its length.
Converse: a line drawn through the mid-point of one side, parallel to another side, bisects the third side.
So if the third side is 10 cm, the mid-segment is 5 cm — a quick way to find lengths without measuring.

Where you'll meet it

Quadrilaterals, all around you

Architecture & frames

Doors, windows and picture frames are rectangles: equal diagonals and four right angles keep them square and rigid. Builders check a frame is "true" by measuring both diagonals — if they are equal, the corners are right angles.

Tiling & floor patterns

Squares, rectangles and rhombi tile a floor with no gaps because four corners meet around a point at 360° — the same angle-sum that every quadrilateral obeys, which is why even irregular four-sided tiles can be arranged to fit.