Quadrilaterals
A quadrilateral is a closed figure with four sides, four vertices and four angles. This Class 8 chapter shows that the four angles of any quadrilateral add up to 360°, classifies the special quadrilaterals, and works out the properties of parallelograms and their special cases — rectangle, rhombus and square.
Learning objectives
- Use the angle sum property of a quadrilateral (360°).
- Classify quadrilaterals by their sides and angles.
- State and apply the properties of a parallelogram.
- Identify how rectangles, rhombuses and squares are special parallelograms.
Key concepts
Angle sum of a quadrilateral
A quadrilateral can be split into two triangles by a diagonal. Each triangle's angles add to 180°, so the four angles of a quadrilateral add to 2 × 180° = 360°. This lets us find a missing angle when the other three are known.
Types of quadrilaterals
A trapezium has one pair of parallel sides. A parallelogram has both pairs of opposite sides parallel. A rhombus is a parallelogram with all sides equal, a rectangle is a parallelogram with all right angles, and a square is both — equal sides and right angles. A kite has two pairs of equal adjacent sides.
Properties of a parallelogram
In a parallelogram, opposite sides are equal and parallel, opposite angles are equal, and adjacent angles are supplementary (add to 180°). The two diagonals bisect each other, meeting at their midpoints.
Special parallelograms
A rectangle adds equal diagonals to the parallelogram properties. A rhombus adds diagonals that bisect each other at right angles. A square, being both, has all of these: equal sides, right angles, and equal diagonals that bisect at 90°.
Important formulas
Angle sum (quadrilateral)
∠A + ∠B + ∠C + ∠D = 360°
Adjacent angles of parallelogram
sum of two adjacent angles = 180°
Angle sum of a polygon
(n − 2) × 180°, for n sides
Key definitions
- Quadrilateral
- A closed figure bounded by four line segments.
- Parallelogram
- A quadrilateral with both pairs of opposite sides parallel.
- Rhombus
- A parallelogram with all four sides equal.
- Trapezium
- A quadrilateral with exactly one pair of parallel sides.
Solved examples
Q1. Three angles of a quadrilateral are 90°, 80° and 100°. Find the fourth.
Solution: Sum is 360°, so the fourth = 360 − (90 + 80 + 100) = 360 − 270 = 90°.
Q2. In a parallelogram one angle is 70°. Find the other three.
Solution: Opposite angle = 70°. Adjacent angles are supplementary: 180 − 70 = 110°. So angles are 70°, 110°, 70°, 110°.
Q3. Why is every square also a rectangle?
Solution: A square has four right angles and opposite sides parallel and equal — exactly the rectangle conditions — so a square is a special rectangle.
Common mistakes to avoid
- Using 180° instead of 360° for the angle sum of a quadrilateral.
- Thinking a trapezium has two pairs of parallel sides (it has exactly one).
- Assuming the diagonals of every parallelogram are equal (only rectangles/squares).
- Believing a rhombus must have right angles — only a square does.
Quadrilaterals — MCQ Quiz
10 questions with instant feedback. Use number keys 1–4 to answer.
The sum of the angles of a quadrilateral is:
Practice questions
Short answer
State the angle sum property of a quadrilateral.
The four interior angles of a quadrilateral add up to 360°.
How are the diagonals of a parallelogram related?
They bisect each other (cross at their midpoints).
What makes a square different from a rectangle?
A square has all four sides equal, while a rectangle only needs opposite sides equal.
Long answer
List the properties of a parallelogram and name its three special cases.
In a parallelogram: opposite sides are equal and parallel, opposite angles are equal, adjacent angles are supplementary, and the diagonals bisect each other. Its special cases are the rectangle (all right angles, equal diagonals), the rhombus (all sides equal, diagonals perpendicular), and the square (both — equal sides, right angles, equal perpendicular-bisecting diagonals).
Show, using triangles, why a quadrilateral's angles sum to 360°.
Draw one diagonal of the quadrilateral. It divides the figure into two triangles. The angles of each triangle sum to 180°, and together these angles make up all four angles of the quadrilateral, so the total is 180° + 180° = 360°.
HOTS (Higher Order Thinking)
Can a quadrilateral have three obtuse angles? Explain.
Yes. Three angles just over 90° (say 100° each = 300°) leave 60° for the fourth, still summing to 360°. It cannot have four obtuse angles, since that would exceed 360°.
The angles of a quadrilateral are in the ratio 1:2:3:4. Find them.
Let the parts be x, 2x, 3x, 4x. Their sum 10x = 360°, so x = 36°. The angles are 36°, 72°, 108° and 144°.
Quick revision
Revision notes
- Quadrilateral angle sum = 360°.
- Parallelogram: opposite sides/angles equal; diagonals bisect each other.
- Rectangle = right angles + equal diagonals; rhombus = equal sides + ⟂ diagonals; square = both.
- Polygon angle sum = (n − 2) × 180°.
Key takeaways
- Every special quadrilateral inherits the parallelogram's properties plus its own.
- A diagonal splits a quadrilateral into two triangles.
- Use 360° to find any missing fourth angle.
Frequently asked questions
Is a square a rhombus?
Yes — a square has all sides equal, so it is a special rhombus (with right angles).
Do the diagonals of every parallelogram bisect each other?
Yes, that is a property of all parallelograms; only some also have equal or perpendicular diagonals.
What is the angle sum of a quadrilateral?
Always 360°, found by splitting it into two triangles.