Quadrilaterals
CBSE Class 9 · Mathematics · Notes, formulas and practice questions
Revise the Class 9 math chapter on quadrilaterals: the angle-sum property, types such as parallelogram, rectangle, rhombus and square, their diagonal properties, conditions that make a quadrilateral a parallelogram, and the mid-point theorem.
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The chapter builds from the simple fact that every quadrilateral has four sides, four vertices and four angles. The angle-sum property states that the four interior angles of any quadrilateral always add up to 360°. This property forms the basis for finding unknown angles and is used repeatedly with parallelogram properties.
Quadrilaterals are classified by how their sides and angles relate. A trapezium has one pair of parallel sides, while a parallelogram has both pairs of opposite sides parallel. Rectangles, rhombuses and squares are special parallelograms, each adding an extra condition: right angles for a rectangle, all sides equal for a rhombus, and both for a square. Learning these definitions clarifies which properties apply.
A central idea is deciding when a quadrilateral is a parallelogram. Several simple conditions work, such as both pairs of opposite sides being equal, or one pair of opposite sides being equal and parallel. This chapter also explores diagonal properties: diagonals of a parallelogram bisect each other, diagonals of a rectangle are equal, and diagonals of a rhombus are perpendicular.
The mid-point theorem connects quadrilateral geometry with triangles. It says the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length, and the converse version lets us prove that a point is a midpoint when a line through one midpoint is parallel to the opposite side. These tools are combined to prove many results.
Key terms
- Quadrilateral
- A closed figure with four straight sides and four angles. The sum of its interior angles is always 360°.
- Angle-sum property
- For any quadrilateral, the four interior angles add up to 360°. Written as A + B + C + D = 360°, where A, B, C, D are the angle measures.
- Trapezium
- A quadrilateral with exactly one pair of parallel sides. In an isosceles trapezium, the non-parallel sides are equal.
- Parallelogram
- A quadrilateral in which both pairs of opposite sides are parallel. Its opposite sides are equal, opposite angles are equal, and its diagonals bisect each other.
- Rectangle
- A parallelogram with one right angle. Since adjacent angles in a parallelogram are supplementary, all four angles become 90°. Its diagonals are equal in length.
- Rhombus
- A parallelogram with all four sides equal. Its diagonals are perpendicular and they bisect each other. They also bisect the vertex angles.
- Square
- A parallelogram that is both a rectangle and a rhombus: all sides equal and all angles 90°. Its diagonals are equal, perpendicular, and bisect each other.
- Mid-point theorem
- In a triangle, if a segment joins the midpoints of two sides, then it is parallel to the third side and half its length. Its converse says: a line through the midpoint of one side, parallel to another side, bisects the third side.
Formula sheet
| What | Formula | Notes |
|---|---|---|
| Angle sum of a quadrilateral | A + B + C + D = 360° | A, B, C and D are the four interior angles of any quadrilateral, measured in degrees. |
| Mid-point theorem | DE ∥ BC and DE = ½BC | In triangle ABC, D is the midpoint of AB and E is the midpoint of AC. The segment DE is then parallel to BC and has length exactly half of BC. |
Practice questions with answers
1. Three angles of a quadrilateral are 80°, 95° and 120°. Find the fourth angle.
Sum of all four angles = 360°. Therefore the fourth angle = 360° − (80° + 95° + 120°) = 360° − 295° = 65°.
2. In a parallelogram, one angle is 62°. Find the measures of the other three angles.
Opposite angles are equal, so the angle opposite the given one is also 62°. Adjacent angles are supplementary, giving 180° − 62° = 118°. Thus the four angles are 62°, 118°, 62° and 118°.
3. State the converse of the mid-point theorem.
If a line is drawn through the midpoint of one side of a triangle, parallel to another side, then it bisects the third side. In triangle ABC, if D is the midpoint of AB and DE is parallel to BC meeting AC at E, then E is the midpoint of AC.
4. In triangle ABC, D and E are the midpoints of AB and AC. If BC = 7.2 cm, find DE.
By the mid-point theorem, the segment joining the midpoints of two sides is half the length of the third side. Therefore DE = ½ × 7.2 cm = 3.6 cm.
5. In a rectangle, one side is 6 cm and a diagonal is 10 cm. Find the other side.
In a rectangle, adjacent sides form a right angle, so the diagonal is the hypotenuse of a right triangle. Using Pythagoras: other side = √(10² − 6²) = √(100 − 36) = √64 = 8 cm.
6. Which single condition about one pair of opposite sides is enough to prove that a quadrilateral is a parallelogram?
If one pair of opposite sides is both equal and parallel, the quadrilateral is a parallelogram. This is a standard criterion: AB = DC and AB ∥ DC imply ABCD is a parallelogram.
7. The angles of a quadrilateral are in the ratio 2 : 3 : 4 : 6. Find the largest angle.
Let the angles be 2x, 3x, 4x and 6x. Their sum is 15x = 360°, so x = 24°. The largest angle is 6 × 24° = 144°.
8. Diagonals of a rhombus are 6 cm and 8 cm. Find the side of the rhombus.
Diagonals of a rhombus bisect each other at right angles, forming four right triangles. Half-diagonals are 3 cm and 4 cm, so side = √(3² + 4²) = √(9 + 16) = √25 = 5 cm.
9. State the diagonal property that a rectangle has but a general parallelogram does not, and the corresponding property of a rhombus.
In every parallelogram the diagonals bisect each other. A rectangle has one further property: its diagonals are also equal in length, so AC = BD. A rhombus instead has diagonals that are perpendicular to each other, and they bisect the angles of the rhombus. A square, being both, has diagonals that are equal, perpendicular and bisect each other.
10. In a rhombus, one diagonal is 12 cm and a side is 10 cm. Find the length of the other diagonal.
The diagonals bisect each other at right angles. Half of the given diagonal is 6 cm. In the right triangle formed, the other half-diagonal is √(10² − 6²) = √(100 − 36) = √64 = 8 cm. Hence the full diagonal is 2 × 8 cm = 16 cm.
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