Understanding Quadrilaterals is an important chapter of Class 8 Mathematics. In this chapter, students learn about quadrilaterals, their different types, angles, sides, diagonals, properties, and angle-sum relationships. This complete guide explains the Understanding Quadrilaterals formulas, properties, important definitions, 15 important solved questions, and exam notes in easy English.
A strong understanding of quadrilaterals is useful for solving geometry questions in Class 8 and also helps students build a foundation for higher-level geometry. The most important ideas include the angle sum property of quadrilaterals, properties of parallelograms, rectangles, squares, rhombuses, trapeziums, kites, and special quadrilaterals.
Understanding Quadrilaterals: Formulas, Properties, Important Questions and Notes
A quadrilateral is a closed two-dimensional figure made up of four line segments. The word quadrilateral comes from the words “quadri,” meaning four, and “lateral,” meaning sides. Therefore, every quadrilateral has four sides, four vertices, and four angles.
What Is a Quadrilateral?
A quadrilateral is a closed polygon having four sides, four vertices, and four interior angles.
If a quadrilateral is named ABCD, then:
- AB, BC, CD and DA are its four sides.
- A, B, C and D are its four vertices.
- ∠A, ∠B, ∠C and ∠D are its four interior angles.
- AC and BD are its diagonals.
Number of vertices = 4
Number of diagonals = 2
Angle Sum Property of a Quadrilateral
The most important formula in this chapter is the angle sum property of a quadrilateral.
If the four angles of a quadrilateral are A, B, C and D, then:
This property can be understood by dividing a quadrilateral into two triangles. Each triangle has an angle sum of 180°. Therefore:
Important Formulas and Properties
1. Sum of Interior Angles
2. Number of Diagonals
3. Regular Polygon Interior Angle Formula
For a regular polygon having n sides:
For a regular polygon, each interior angle is:
4. Exterior Angle of a Regular Polygon
5. Sum of Exterior Angles
6. Parallelogram Properties
Opposite angles are equal.
Adjacent angles are supplementary.
Diagonals bisect each other.
7. Rectangle Properties
Opposite sides are equal and parallel.
Diagonals are equal.
Diagonals bisect each other.
8. Square Properties
All four angles = 90°.
Opposite sides are parallel.
Diagonals are equal.
Diagonals bisect each other at right angles.
9. Rhombus Properties
Opposite angles are equal.
Opposite sides are parallel.
Diagonals bisect each other at right angles.
10. Trapezium Properties
In an isosceles trapezium, the non-parallel sides are equal and the base angles are equal.
11. Kite Properties
Its diagonals have special properties, including perpendicular intersection in a kite.
Types of Quadrilaterals
There are different types of quadrilaterals based on their sides, angles, and parallel lines.
Parallelogram
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
- Opposite sides are equal.
- Opposite sides are parallel.
- Opposite angles are equal.
- Adjacent angles add up to 180°.
- Diagonals bisect each other.
Rectangle
A rectangle is a parallelogram in which all four angles are right angles.
- All angles are 90°.
- Opposite sides are equal.
- Opposite sides are parallel.
- Diagonals are equal.
- Diagonals bisect each other.
Square
A square is a special quadrilateral having four equal sides and four right angles.
- All sides are equal.
- All angles are 90°.
- Opposite sides are parallel.
- Diagonals are equal.
- Diagonals are perpendicular.
- Diagonals bisect each other.
Rhombus
A rhombus is a parallelogram in which all four sides are equal.
- All sides are equal.
- Opposite sides are parallel.
- Opposite angles are equal.
- Diagonals bisect each other at right angles.
Trapezium
A trapezium is a quadrilateral having one pair of parallel sides.
Kite
A kite is a quadrilateral having two pairs of adjacent equal sides.
Difference Between Important Quadrilaterals
| Quadrilateral | Main Property |
|---|---|
| Parallelogram | Both pairs of opposite sides are parallel. |
| Rectangle | All angles are 90°. |
| Square | All sides are equal and all angles are 90°. |
| Rhombus | All four sides are equal. |
| Trapezium | One pair of opposite sides is parallel. |
| Kite | Two pairs of adjacent sides are equal. |
Polygon and Its Important Formula
A polygon is a closed plane figure made up of three or more line segments. A triangle has three sides, a quadrilateral has four sides, a pentagon has five sides, and so on.
For example, for a pentagon:
For a hexagon:
Exterior Angles of a Polygon
The exterior angle is formed when one side of a polygon is extended.
For a regular polygon, all exterior angles are equal.
15 Important Questions with Solutions
Question 1: Find the fourth angle of a quadrilateral if three angles are 80°, 90° and 100°.
Sum of angles of a quadrilateral = 360°
Fourth angle = 360° − (80° + 90° + 100°)
= 360° − 270°
= 90°
Answer: 90°
Question 2: Three angles of a quadrilateral are 70°, 80° and 110°. Find the fourth angle.
Fourth angle = 360° − (70° + 80° + 110°)
= 360° − 260°
= 100°
Answer: 100°
Question 3: If all four angles of a quadrilateral are equal, find each angle.
Sum of angles = 360°
Each angle = 360°/4
= 90°
Answer: Each angle is 90°.
Question 4: The angles of a quadrilateral are x°, 2x°, 3x° and 4x°. Find x.
x + 2x + 3x + 4x = 360°
10x = 360°
x = 36°
Answer: x = 36°
Question 5: Find the angles of a quadrilateral if they are x°, x°, 2x° and 2x°.
x + x + 2x + 2x = 360°
6x = 360°
x = 60°
Therefore, the angles are:
60°, 60°, 120°, 120°.
Answer: 60°, 60°, 120° and 120°.
Question 6: One angle of a parallelogram is 70°. Find its opposite angle.
Opposite angles of a parallelogram are equal.
Therefore, opposite angle = 70°.
Answer: 70°
Question 7: One angle of a parallelogram is 65°. Find its adjacent angle.
Adjacent angles of a parallelogram are supplementary.
Adjacent angle = 180° − 65°
= 115°
Answer: 115°
Question 8: The diagonals of a parallelogram intersect at O. If AO = 7 cm, find OC.
The diagonals of a parallelogram bisect each other.
Therefore, AO = OC.
OC = 7 cm
Answer: OC = 7 cm
Question 9: All four angles of a rectangle are what?
A rectangle has four right angles.
Each right angle = 90°.
Answer: All four angles are 90°.
Question 10: A square has a side of 8 cm. What is the length of each side?
All four sides of a square are equal.
Therefore, each side = 8 cm.
Answer: Each side is 8 cm.
Question 11: How many diagonals does a quadrilateral have?
A quadrilateral has four vertices and two diagonals connecting opposite vertices.
Answer: 2 diagonals.
Question 12: Find the sum of the interior angles of a pentagon.
Formula:
(n − 2) × 180°
For a pentagon, n = 5.
(5 − 2) × 180°
= 3 × 180°
= 540°
Answer: 540°
Question 13: Find each exterior angle of a regular hexagon.
Formula:
Each exterior angle = 360°/n
For a hexagon, n = 6.
360°/6 = 60°
Answer: 60°
Question 14: The angles of a quadrilateral are 90°, 90°, 80° and x°. Find x.
90° + 90° + 80° + x = 360°
260° + x = 360°
x = 360° − 260°
x = 100°
Answer: x = 100°
Question 15: The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. Find all the angles.
Let the angles be x, 2x, 3x and 4x.
x + 2x + 3x + 4x = 360°
10x = 360°
x = 36°
Therefore:
x = 36°
2x = 72°
3x = 108°
4x = 144°
Answer: 36°, 72°, 108° and 144°.
Important Notes for Exams
- The sum of the interior angles of a quadrilateral is always 360°.
- A quadrilateral has 4 sides, 4 vertices and 2 diagonals.
- Opposite sides of a parallelogram are equal and parallel.
- Opposite angles of a parallelogram are equal.
- Adjacent angles of a parallelogram are supplementary.
- The diagonals of a parallelogram bisect each other.
- All angles of a rectangle are 90°.
- All sides of a square are equal.
- A rhombus has four equal sides.
- A trapezium has one pair of parallel sides.
- A kite has two pairs of adjacent equal sides.
- The sum of exterior angles of any polygon is 360°.
Common Mistakes Students Should Avoid
- Using 180° instead of 360° for the angle sum of a quadrilateral.
- Confusing opposite angles with adjacent angles.
- Forgetting that all four angles of a rectangle are 90°.
- Assuming every quadrilateral has equal sides.
- Confusing a rhombus with a square.
- Forgetting that a parallelogram has two pairs of parallel opposite sides.
- Using the regular polygon formula when the polygon is not regular.
- Making calculation mistakes while solving angle equations.
Quick Revision Table
| Shape | Important Properties |
|---|---|
| Quadrilateral | 4 sides, 4 angles, 2 diagonals; angle sum = 360°. |
| Parallelogram | Opposite sides parallel and equal; opposite angles equal. |
| Rectangle | Four right angles; opposite sides equal and parallel. |
| Square | Four equal sides and four right angles. |
| Rhombus | Four equal sides; opposite angles equal. |
| Trapezium | One pair of parallel sides. |
| Kite | Two pairs of adjacent equal sides. |
How to Solve Quadrilateral Angle Questions
When you see an angle question involving a quadrilateral, the first thing to remember is the 360° angle sum property.
For example, if three angles are known and the fourth angle is x, write:
Then simplify the equation and find x.
If the question involves a parallelogram, use the special properties of a parallelogram. Opposite angles are equal, while adjacent angles add up to 180°.
Understanding Regular Polygons
A regular polygon has all sides equal and all interior angles equal. Examples include a regular triangle, regular square, regular pentagon, regular hexagon, and so on.
The general formula for the sum of interior angles of an n-sided polygon is:
If the polygon is regular, divide the sum by the number of sides to find each interior angle.
Similarly:
Frequently Asked Questions
What is a quadrilateral?
A quadrilateral is a closed polygon having four sides, four vertices, and four interior angles.
What is the sum of angles of a quadrilateral?
The sum of the four interior angles of every quadrilateral is 360°.
How many diagonals does a quadrilateral have?
A quadrilateral has two diagonals.
What are the properties of a parallelogram?
Opposite sides are equal and parallel, opposite angles are equal, adjacent angles are supplementary, and diagonals bisect each other.
What is the difference between a square and a rectangle?
Both have four right angles, but a square has all four sides equal, while a rectangle has only opposite sides equal.
What is a rhombus?
A rhombus is a quadrilateral in which all four sides are equal. Its opposite sides are parallel and opposite angles are equal.
What is a trapezium?
A trapezium is a quadrilateral having one pair of parallel sides.
What is the formula for the sum of angles of a polygon?
For an n-sided polygon, the sum of interior angles is (n − 2) × 180°.
What is the exterior angle formula for a regular polygon?
Each exterior angle of a regular polygon is 360° divided by the number of sides, or 360°/n.
Conclusion
Understanding Quadrilaterals is an important Class 8 Maths chapter that introduces students to different types of quadrilaterals and their properties. The most important concept is that the sum of the interior angles of a quadrilateral is 360°.
Students should also remember the properties of parallelograms, rectangles, squares, rhombuses, trapeziums, and kites. For polygon questions, the formula (n − 2) × 180° is very useful for finding the sum of interior angles.
Regular practice of Class 8 Understanding Quadrilaterals important questions can improve calculation speed and geometric reasoning. Before solving any problem, identify the shape and then apply the correct property or formula.
Disclaimer
This article is provided for educational and revision purposes. The explanations, formulas, examples, and questions are intended to help Class 8 students understand the chapter. Students should also follow their school textbook, teacher's instructions, and prescribed syllabus for examination preparation.