Chapter focus
Quadrilaterals are four-sided polygons. Classifying them by parallel sides, equal sides and angles lets us identify their properties and solve missing-angle problems.
These notes give original explanations and examples for revision. Use your assigned Ganita Prakash edition for the complete diagrams, activities and exercises.
1. Angle sum of a quadrilateral
A diagonal divides a quadrilateral into two triangles. Each triangle has an angle sum of 180°, so the four interior angles together add to 180° + 180° = 360°. This applies to every simple quadrilateral.
Worked example
A quadrilateral has angles 74°, 106° and 91°. The fourth is 360° − (74° + 106° + 91°) = 360° − 271° = 89°. Check that the four angles total 360°.
2. Important quadrilaterals and their properties
- Parallelogram: both pairs of opposite sides are parallel. Opposite sides and opposite angles are equal; adjacent angles are supplementary; diagonals bisect each other.
- Rectangle: a parallelogram with four right angles. Opposite sides are equal and parallel; diagonals are equal and bisect each other.
- Rhombus: a parallelogram with four equal sides. Opposite angles are equal; diagonals bisect each other at right angles.
- Square: has four equal sides and four right angles, so it has the properties of both a rectangle and a rhombus.
- Kite: has two pairs of adjacent equal sides. One pair of opposite angles is equal; its diagonals meet at right angles, and one diagonal bisects the other.
- Trapezium: a quadrilateral with one pair of opposite sides parallel (in the common Indian school definition).
These categories can overlap. Every square is both a rectangle and a rhombus, but a rectangle need not have all four sides equal.
3. Use properties to find unknown angles
In a parallelogram, adjacent interior angles lie on the same side of a transversal between parallel lines, so they add to 180°. Opposite angles are equal.
Worked example
One angle of a parallelogram is 68°. The adjacent angle is 180° − 68° = 112°. The opposite angle is 68°, and the remaining angle is 112°. The four angles are 68°, 112°, 68°, 112° and total 360°.
Do not assume a quadrilateral is a parallelogram from its appearance in a sketch. Use the stated parallel or equal-side information to justify the property.
4. Compare rectangle, rhombus and square
Four right angles; opposite sides equal; diagonals equal.
Four equal sides; opposite angles equal; diagonals perpendicular.
Four equal sides and four right angles; it has both sets of properties.
A useful classification question is: “What extra condition turns this shape into a square?” A rectangle needs all four sides equal; a rhombus needs one right angle (which forces all angles to be right angles).
Common errors to avoid
- Equal sides do not automatically mean right angles: a rhombus may be slanted.
- Equal diagonals alone do not prove a quadrilateral is a rectangle.
- Adjacent angles of a parallelogram add to 180°; opposite angles are equal.
- Label diagrams clearly and cite the property used in each step.
Practice: include a reason
1. Find the fourth angle if the other angles are 72°, 98° and 110°.
Answer: 360° − (72° + 98° + 110°) = 80°.
2. One angle in a parallelogram is 127°. Find the other three.
Answer: The opposite angle is 127°. Each adjacent angle is 180° − 127° = 53°. The angles are 127°, 53°, 127°, 53°.
3. A quadrilateral has four equal sides and one right angle. What shape is it? Explain.
Answer: It is a square. A rhombus has equal sides; adjacent angles are supplementary, so if one is 90°, its adjacent angle is also 90°, and the opposite angles are equal. All four angles are right angles.
4. Name two properties shared by a rectangle and a parallelogram.
Answer: Both have two pairs of parallel opposite sides, equal opposite sides, and diagonals that bisect each other.
5. Three angles of a quadrilateral are 90°, 90° and 90°. What must the fourth angle be?
Answer: 360° − 270° = 90°. All four are right angles, so the quadrilateral is a rectangle; more information is needed to conclude it is a square.
Chapter summary
Use parallelism, side equality, angle relationships and diagonal properties to classify quadrilaterals. The interior-angle sum is 360°. Distinguish a rectangle, rhombus and square by checking which defining conditions are given.