Quadrilaterals
A quadrilateral is a closed figure with four sides. This Class 9 chapter covers the angle sum property of quadrilaterals, the family of special quadrilaterals, the key properties of a parallelogram and the conditions that make a quadrilateral one, and the very useful mid-point theorem. The parallelogram results are favourites in board proofs.
Learning objectives
- Use the angle sum property of a quadrilateral.
- Identify the special quadrilaterals and their properties.
- State and apply the properties of a parallelogram.
- Recognise the conditions for a quadrilateral to be a parallelogram.
- Apply the mid-point theorem.
Key concepts
Angle sum property
The sum of the four interior angles of any quadrilateral is 360°. This follows from splitting the quadrilateral into two triangles, each contributing 180°.
Special quadrilaterals
Quadrilaterals include the parallelogram, rectangle, rhombus, square and trapezium. A square is both a rectangle and a rhombus; a rectangle and a rhombus are special parallelograms.
Properties of a parallelogram
In a parallelogram: opposite sides are equal and parallel, opposite angles are equal, consecutive angles are supplementary, and the diagonals bisect each other.
Mid-point theorem
The line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it. Its converse is also true and is often used in proofs.
Important formulas
Angle sum of a quadrilateral
∠A + ∠B + ∠C + ∠D = 360°
Parallelogram diagonals
the diagonals bisect each other
Mid-point theorem
segment joining mid-points = ½ × third side, and parallel to it
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 angle.
Solution: Fourth angle = 360° − (90° + 80° + 100°) = 360° − 270° = 90°.
Q2. In a parallelogram, one angle is 110°. Find the other three angles.
Solution: Opposite angle = 110°. Consecutive angles are supplementary, so the other two = 180° − 110° = 70° each. Angles: 110°, 70°, 110°, 70°.
Q3. In a triangle of side 12 cm, what is the length of the segment joining the mid-points of the other two sides relative to that side?
Solution: By the mid-point theorem, this segment is half of 12 cm = 6 cm and is parallel to that side.
Common mistakes to avoid
- Taking the angle sum of a quadrilateral as 180° instead of 360°.
- Assuming every parallelogram has equal diagonals — only rectangles and squares do.
- Confusing a trapezium (one pair parallel) with a parallelogram (two pairs).
- Forgetting the 'half' in the mid-point theorem.
Quadrilaterals — MCQ Quiz
10 questions with instant feedback. Use number keys 1–4 to answer.
The sum of the interior angles of a quadrilateral is:
Practice questions
Short answer
What is the angle sum of a quadrilateral?
360°.
Name one property that a rhombus has but a general parallelogram may not.
All four sides are equal (its diagonals are also perpendicular).
State the 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.
Long answer
Prove that in a parallelogram the opposite angles are equal (outline).
In parallelogram ABCD, AB ∥ DC and AD ∥ BC. Using the alternate-angle property with diagonal AC, triangles ABC and CDA are congruent (ASA). Hence ∠B = ∠D, and similarly ∠A = ∠C — opposite angles are equal.
List the properties of a parallelogram.
Opposite sides are equal and parallel; opposite angles are equal; consecutive angles are supplementary; the diagonals bisect each other. A rectangle adds equal diagonals; a rhombus adds equal sides with perpendicular diagonals; a square has both.
HOTS (Higher Order Thinking)
If the diagonals of a quadrilateral bisect each other, what type of quadrilateral is it?
It must be a parallelogram, because diagonals bisecting each other is a sufficient condition for a parallelogram.
The diagonals of a quadrilateral are equal and bisect each other at right angles. What is it?
It is a square — equal diagonals (rectangle), bisecting at right angles (rhombus), so both conditions give a square.
Quick revision
Revision notes
- Angle sum of a quadrilateral = 360°.
- Parallelogram: opposite sides/angles equal, diagonals bisect each other.
- Rectangle, rhombus, square are special parallelograms; trapezium has one parallel pair.
- Mid-point theorem: joining segment = ½ third side and parallel to it.
Key takeaways
- Use the 360° rule to find a missing angle.
- Know which special parallelogram has which extra property.
- The mid-point theorem and its converse are powerful in proofs.
Frequently asked questions
What is the angle sum of a quadrilateral?
The four interior angles of any quadrilateral add up to 360°.
What are the properties of a parallelogram?
Opposite sides are equal and parallel, opposite angles are equal, consecutive angles are supplementary, and the diagonals bisect each other.
What is the mid-point theorem?
The line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it.