Properties of Quadrilaterals
A quadrilateral is a closed shape with four sides, four vertices, and four angles. The sum of all interior angles of any quadrilateral is 360°. In Class 5, students learn the properties of special quadrilaterals: parallelogram, rectangle, square, rhombus, and trapezium.
Understanding these properties helps in identifying shapes, solving geometry problems, and calculating perimeter and area.
What is Properties of Quadrilaterals - Class 5 Maths (Geometry)?
A quadrilateral is a polygon with exactly four sides. Special quadrilaterals have additional properties related to their sides, angles, and diagonals.
| Shape | Key Properties |
|---|---|
| Parallelogram | Opposite sides parallel and equal; opposite angles equal; diagonals bisect each other |
| Rectangle | All angles 90°; opposite sides equal; diagonals equal and bisect each other |
| Square | All sides equal; all angles 90°; diagonals equal and bisect at right angles |
| Rhombus | All sides equal; opposite angles equal; diagonals bisect at right angles |
| Trapezium | One pair of opposite sides parallel |
Properties of Quadrilaterals Formula
Sum of angles of a quadrilateral = 360°
Types and Properties
Hierarchy of quadrilaterals:
- A square is a special rectangle (all sides equal) AND a special rhombus (all angles 90°).
- A rectangle is a special parallelogram (all angles 90°).
- A rhombus is a special parallelogram (all sides equal).
- A parallelogram is a special trapezium (both pairs of sides parallel).
Every square is a rectangle, rhombus, and parallelogram. But not every rectangle is a square.
Solved Examples
Example 1: Example 1: Finding the fourth angle
Problem: Three angles of a quadrilateral are 80°, 100°, and 90°. Find the fourth angle.
Solution:
Sum of all angles = 360°
Fourth angle = 360° − (80° + 100° + 90°) = 360° − 270° = 90°
Answer: The fourth angle is 90°.
Example 2: Example 2: Identifying a parallelogram
Problem: A quadrilateral has opposite sides of 5 cm and 8 cm. The opposite sides are parallel. What type is it?
Solution:
Opposite sides parallel and equal → Parallelogram.
Since the sides are not all equal, it is not a rhombus or square. Since we do not know if all angles are 90°, we cannot say it is a rectangle.
Example 3: Example 3: Properties of a rectangle
Problem: The diagonals of a rectangle are 10 cm each. The diagonals intersect at point O. Find OA if A is a vertex.
Solution:
In a rectangle, diagonals are equal and bisect each other.
Each half = 10/2 = 5 cm.
Answer: OA = 5 cm.
Example 4: Example 4: Properties of a rhombus
Problem: A rhombus has one angle of 60°. Find all four angles.
Solution:
In a rhombus, opposite angles are equal.
If one angle = 60°, the opposite angle = 60°.
Adjacent angles are supplementary: 180° − 60° = 120°.
Answer: The four angles are 60°, 120°, 60°, 120°.
Example 5: Example 5: Square properties
Problem: The diagonal of a square is 10 cm. Do the diagonals bisect at right angles?
Solution:
Yes. In a square, the diagonals are equal, bisect each other, and meet at right angles (90°).
Each half-diagonal = 10/2 = 5 cm.
Example 6: Example 6: Identifying shapes from properties
Problem: A quadrilateral has all sides equal and all angles 90°. What is it?
Solution:
All sides equal → could be rhombus or square.
All angles 90° → must be a rectangle.
All sides equal + all angles 90° = Square.
Example 7: Example 7: Trapezium
Problem: In a trapezium ABCD, AB is parallel to DC. If angle A = 70° and angle B = 110°, find angles C and D.
Solution:
In a trapezium with AB ∥ DC:
Angle A + Angle D = 180° (co-interior angles) → D = 180° − 70° = 110°
Angle B + Angle C = 180° → C = 180° − 110° = 70°
Answer: C = 70°, D = 110°.
Example 8: Example 8: Perimeter of a rhombus
Problem: A rhombus has a side of 9 cm. Find its perimeter.
Solution:
All sides of a rhombus are equal.
Perimeter = 4 × side = 4 × 9 = 36 cm.
Answer: Perimeter = 36 cm.
Example 9: Example 9: Word problem
Problem: Dev is making a photo frame in the shape of a parallelogram. Two angles are 65° each. Find the other two angles.
Solution:
In a parallelogram, opposite angles are equal and consecutive angles are supplementary.
Other two angles = 180° − 65° = 115° each.
Answer: The four angles are 65°, 115°, 65°, 115°.
Key Points to Remember
- Sum of all angles of a quadrilateral = 360°.
- Parallelogram: Opposite sides parallel and equal; opposite angles equal.
- Rectangle: Parallelogram with all angles 90°; diagonals are equal.
- Square: Rectangle with all sides equal; diagonals bisect at 90°.
- Rhombus: Parallelogram with all sides equal; diagonals bisect at 90°.
- Trapezium: Exactly one pair of parallel sides.
- Every square is a rectangle, but not every rectangle is a square.
- Diagonals of parallelograms bisect each other; in rectangles and squares they are equal.
Practice Problems
- Three angles of a quadrilateral are 75°, 85°, and 105°. Find the fourth angle.
- A parallelogram has angles of 70° and 110°. Verify that opposite angles are equal.
- The sides of a rhombus are 12 cm. Find its perimeter.
- Is every square a rhombus? Is every rhombus a square? Explain.
- A rectangle has a length of 15 cm and a width of 8 cm. Find the perimeter.
- A trapezium has parallel sides of 10 cm and 6 cm and the other two sides of 5 cm each. Find its perimeter.
- In a parallelogram, one angle is 55°. Find all four angles.
- Name a quadrilateral whose diagonals are equal and bisect each other at right angles.
Frequently Asked Questions
Q1. What is the sum of angles in a quadrilateral?
The sum of all four interior angles of any quadrilateral is 360°. This applies to all quadrilaterals — regular, irregular, convex, or concave.
Q2. What is the difference between a rhombus and a square?
Both have all four sides equal. A square has all angles equal to 90°, while a rhombus can have angles other than 90°. A square is a special case of a rhombus where all angles are right angles.
Q3. Is a rectangle a parallelogram?
Yes. A rectangle has two pairs of parallel sides and opposite sides equal, which are properties of a parallelogram. A rectangle is a special parallelogram with all angles 90°.
Q4. What makes a trapezium different from a parallelogram?
A trapezium has exactly one pair of parallel sides. A parallelogram has two pairs of parallel sides. So a parallelogram is a more specific shape than a trapezium.
Q5. Do diagonals of all quadrilaterals bisect each other?
No. Diagonals bisect each other only in parallelograms (including rectangles, squares, and rhombuses). In a general quadrilateral or trapezium, diagonals do not necessarily bisect each other.
Q6. Which quadrilateral has diagonals that are perpendicular?
In a rhombus and a square, the diagonals bisect each other at right angles (90°). In a rectangle that is not a square, the diagonals are equal but not perpendicular.
Q7. Can a quadrilateral have all four angles different?
Yes. An irregular quadrilateral can have all four angles different, as long as they add up to 360°. For example: 60°, 80°, 100°, 120°.
Q8. Is this topic important for Class 5 exams?
Yes. Properties of quadrilaterals — identifying types, finding missing angles, and understanding relationships between shapes — are commonly tested in Class 5 CBSE Maths exams.
Related Topics
- Quadrilaterals (Grade 5)
- Types of Triangles (Grade 5)
- Lines, Line Segments and Rays
- Parallel and Perpendicular Lines
- Angles (Grade 5)
- Angle Sum Property of Triangle
- Circles (Grade 5)
- Symmetry (Grade 5)
- Reflection and Rotation
- Nets of 3D Shapes (Grade 5)
- Views of 3D Shapes (Grade 5)
- Complementary and Supplementary Angles
