Understanding Quadrilaterals

Understanding Quadrilaterals: Complete Guide for Class 8 CBSE with Properties, Formulas & Practice Problems

Introduction to Quadrilaterals

A quadrilateral is a closed figure bounded by four line segments, where each endpoint connects to exactly two other segments. The term comes from "quad" (four) and "lateral" (sides). Quadrilaterals are fundamental two-dimensional shapes in geometry that students encounter frequently in Class 8 mathematics and beyond.

Components of a Quadrilateral

Every quadrilateral consists of:

  • Four sides (line segments forming the boundary)
  • Four vertices (points where sides meet)
  • Four interior angles (formed at each vertex)
  • Two diagonals (line segments joining non-adjacent vertices)

Understanding these components is essential for analyzing quadrilateral properties and solving geometric problems effectively.

Essential Properties Every Student Must Memorize

Universal Quadrilateral Properties

  1. Angle Sum Property: The sum of all interior angles of any quadrilateral equals 360° (or 4 right angles)
  2. Number of diagonals: Every quadrilateral has exactly 2 diagonals
  3. Adjacent sides: Sides sharing a common vertex are adjacent
  4. Opposite sides: Sides that don't share a vertex are opposite
  5. Adjacent angles: Angles sharing a common side are adjacent
  6. Opposite angles: Angles that don't share a side are opposite

Convex vs. Concave Quadrilaterals

  • Convex quadrilateral: All interior angles are less than 180°; any line segment joining two points inside lies completely within the figure
  • Concave quadrilateral: At least one interior angle exceeds 180°; the boundary crosses itself

How to Prove the Sum of Interior Angles Equals 360 Degrees

Theorem: Interior Angle Sum Property

Statement: The sum of the interior angles of any quadrilateral is 360° or 4 right angles.

Proof Method

Given: Quadrilateral ABCD

To Prove: ∠A + ∠B + ∠C + ∠D = 360°

Construction: Draw diagonal AC

Proof:

  1. Diagonal AC divides quadrilateral ABCD into two triangles: △ABC and △ACD
  1. We know that the sum of angles in any triangle = 180°
  1. In △ABC:
    • ∠1 + ∠4 + ∠B = 180° ... (i)
  1. In △ACD:
    • ∠2 + ∠3 + ∠D = 180° ... (ii)
  1. Adding equations (i) and (ii):
    • (∠1 + ∠4 + ∠B) + (∠2 + ∠3 + ∠D) = 180° + 180°
    • (∠1 + ∠2) + ∠B + (∠3 + ∠4) + ∠D = 360°
  1. Since ∠1 + ∠2 = ∠A and ∠3 + ∠4 = ∠C:
    • ∠A + ∠B + ∠C + ∠D = 360°

Hence Proved.

This fundamental property applies to all quadrilaterals regardless of shape or size, making it one of the most important concepts in quadrilateral geometry.

Types of Quadrilaterals: Complete Classification

1. Trapezium (Trapezoid)

Definition: A quadrilateral with exactly one pair of parallel sides.

Properties:

  • One pair of opposite sides is parallel (called bases)
  • The other pair of sides is non-parallel (called legs)
  • Sum of adjacent angles on non-parallel sides = 180°

Isosceles Trapezium: A trapezium with non-parallel sides equal in length and base angles equal.

2. Parallelogram

Definition: A quadrilateral with both pairs of opposite sides parallel.

Essential Properties:

  • Opposite sides are equal: AB = DC, AD = BC
  • Opposite angles are equal: ∠A = ∠C, ∠B = ∠D
  • Diagonals bisect each other
  • Adjacent angles are supplementary: ∠A + ∠B = 180°
  • Each diagonal divides it into two congruent triangles

3. Rhombus

Definition: A parallelogram with all four sides equal in length.

Essential Properties:

  • All sides are equal: AB = BC = CD = DA
  • Opposite angles are equal
  • Diagonals bisect each other at right angles (90°)
  • Diagonals bisect the vertex angles
  • Diagonals are generally NOT equal in length

4. Rectangle

Definition: A parallelogram with all angles equal to 90°.

Essential Properties:

  • All angles are right angles (90°)
  • Opposite sides are equal
  • Diagonals are equal in length
  • Diagonals bisect each other (but NOT at right angles)

5. Square

Definition: A rectangle with all sides equal OR a rhombus with all angles equal to 90°.

Essential Properties:

  • All sides are equal
  • All angles are right angles (90°)
  • Diagonals are equal in length
  • Diagonals bisect each other at right angles
  • Diagonals bisect the vertex angles (forming 45° angles)

6. Kite

Definition: A quadrilateral with exactly two pairs of consecutive sides equal.

Essential Properties:

  • Two pairs of adjacent sides are equal
  • One pair of opposite angles is equal (where unequal sides meet)
  • Diagonals intersect at right angles
  • One diagonal bisects the other

Differences Between Parallelogram, Rhombus, Rectangle, and Square

Comprehensive Comparison Table

Property Parallelogram Rhombus Rectangle Square
Opposite sides parallel Yes Yes Yes Yes
Opposite sides equal Yes Yes Yes Yes
All sides equal No Yes No Yes
Opposite angles equal Yes Yes Yes Yes
All angles 90° No No Yes Yes
Diagonals equal No No Yes Yes
Diagonals bisect each other Yes Yes Yes Yes
Diagonals perpendicular No Yes No Yes
Diagonals bisect angles No Yes No Yes

Hierarchical Relationships

Understanding the relationships between these quadrilaterals helps in problem-solving:

  • Every square is:
    • A rectangle (all angles are 90°)
    • A rhombus (all sides are equal)
    • A parallelogram (opposite sides parallel)
    • A trapezium (at least one pair of parallel sides)
  • Every rectangle is:
    • A parallelogram
    • A trapezium
    • But NOT necessarily a square or rhombus
  • Every rhombus is:
    • A parallelogram
    • A trapezium
    • But NOT necessarily a square or rectangle
  • Every parallelogram is:
    • A trapezium
    • But NOT necessarily a rhombus, rectangle, or square

Polygon Angle Properties

For Regular Polygons with n sides

These formulas apply to regular polygons (all sides and angles equal):

1. Sum of Interior Angles

  • Formula: (n - 2) × 180° or (2n - 4) × 90°
  • Example: For hexagon (n=6): (6-2) × 180° = 720°

2. Each Interior Angle of Regular Polygon

  • Formula: [(2n - 4) × 90°] / n
  • Simplified: [(n - 2) × 180°] / n
  • Example: For regular hexagon: (4 × 90°)/6 = 120°

3. Sum of Exterior Angles

  • Formula: Always 360° (for any polygon)
  • This is constant regardless of the number of sides

4. Each Exterior Angle of Regular Polygon

  • Formula: 360° / n
  • Example: For regular pentagon: 360°/5 = 72°

5. Relationship Between Interior and Exterior Angles

  • Interior angle + Exterior angle = 180° (linear pair)

Number of Sides Calculation

If given an exterior angle, you can find the number of sides:

  • n = 360° / (exterior angle)

If given an interior angle:

  • First find exterior angle: 180° - interior angle
  • Then: n = 360° / (exterior angle)

Area and Perimeter Formulas: Consolidated Reference Table

Quadrilateral Area Formula Perimeter Formula Notes
General Quadrilateral Varies by type Sum of all four sides Use specific formulas below
Parallelogram Base × Height
OR
b × h		 2(a + b)
where a, b are adjacent sides		 Height is perpendicular distance between parallel sides
Rhombus ½ × d₁ × d₂
OR
Base × Height		 4a
where a = side length		 d₁, d₂ are diagonals;
All sides equal
Rectangle Length × Width
OR
l × w		 2(l + w) All angles = 90°
Square Side²
OR
a²		 4a
where a = side length		 All sides equal;
All angles = 90°
Trapezium ½ × (Sum of parallel sides) × Height
OR
½ × (a + b) × h		 Sum of all four sides a, b are parallel sides;
h is perpendicular height
Kite ½ × d₁ × d₂ 2(a + b)
where a, b are pairs of equal sides		 d₁, d₂ are diagonals

Important Notes on Formulas

  1. Height vs. Side: Height is always the perpendicular distance between parallel sides, which may differ from the slant side length
  1. Diagonal Formulas: For rectangles and squares, diagonal length can be calculated using Pythagoras theorem
  1. Unit Consistency: Always ensure all measurements are in the same unit before calculation

Examples with Step-by-Step Solutions

Example 1: Finding Missing Angle in a Quadrilateral

Problem: Three angles of a quadrilateral are 54°, 80°, and 116°. Find the fourth angle.

Solution:

  • Sum of angles in quadrilateral = 360°
  • Let fourth angle = x°
  • 54° + 80° + 116° + x° = 360°
  • 250° + x° = 360°
  • x° = 360° - 250°
  • x° = 110°

Answer: The fourth angle is 110°.

Example 2: Finding Angles in a Parallelogram

Problem: In parallelogram ABCD, if ∠A = 70°, find all other angles.

Solution:

Step 1: Use the property that opposite angles are equal

  • ∠C = ∠A = 70°

Step 2: Use the property that adjacent angles are supplementary

  • ∠A + ∠B = 180°
  • 70° + ∠B = 180°
  • ∠B = 110°

Step 3: Since opposite angles are equal

  • ∠D = ∠B = 110°

Answer: ∠A = 70°, ∠B = 110°, ∠C = 70°, ∠D = 110°

Example 3: Finding Number of Sides from Exterior Angle

Problem: Each exterior angle of a regular polygon measures 72°. How many sides does it have?

Solution:

  • Formula: n = 360° / (exterior angle)
  • n = 360° / 72°
  • n = 5

Answer: The polygon has 5 sides (it's a regular pentagon).

Example 4: Interior Angle of Regular Polygon

Problem: Find each interior angle of a regular hexagon.

Solution:

Method 1 (Direct formula):

  • Each interior angle = [(n - 2) × 180°] / n
  • For hexagon, n = 6
  • Each interior angle = [(6 - 2) × 180°] / 6
  • = (4 × 180°) / 6
  • = 720° / 6
  • = 120°

Method 2 (Using exterior angle):

  • Each exterior angle = 360° / 6 = 60°
  • Each interior angle = 180° - 60° = 120°

Answer: Each interior angle of a regular hexagon is 120°.

Example 5: Area Calculation

Problem: Find the area of a rhombus whose diagonals are 8 cm and 6 cm.

Solution:

  • Area of rhombus = ½ × d₁ × d₂
  • Area = ½ × 8 × 6
  • Area = ½ × 48
  • Area = 24 cm²

Answer: The area of the rhombus is 24 cm².

Example 6: Perimeter of Parallelogram

Problem: The perimeter of a parallelogram is 150 cm. One side is 10 cm greater than the other. Find the lengths of all sides.

Solution:

Step 1: Let shorter side = x cm

  • Then longer side = (x + 10) cm

Step 2: Perimeter formula for parallelogram

  • Perimeter = 2(sum of adjacent sides)
  • 150 = 2[x + (x + 10)]
  • 150 = 2(2x + 10)
  • 150 = 4x + 20
  • 130 = 4x
  • x = 32.5 cm

Step 3: Find both sides

  • Shorter side = 32.5 cm
  • Longer side = 32.5 + 10 = 42.5 cm

Step 4: All four sides

  • Two sides of 32.5 cm each
  • Two sides of 42.5 cm each

Answer: The sides are 32.5 cm, 42.5 cm, 32.5 cm, and 42.5 cm.

Example 7: Angles in Ratio Problem

Problem: The four angles of a quadrilateral are in the ratio 2:3:5:8. Find all angles.

Solution:

Step 1: Let the common ratio be x

  • Angles are 2x°, 3x°, 5x°, 8x°

Step 2: Sum of angles = 360°

  • 2x + 3x + 5x + 8x = 360
  • 18x = 360
  • x = 20

Step 3: Calculate each angle

  • First angle = 2 × 20 = 40°
  • Second angle = 3 × 20 = 60°
  • Third angle = 5 × 20 = 100°
  • Fourth angle = 8 × 20 = 160°

Verification: 40° + 60° + 100° + 160° = 360° ✓

Answer: The angles are 40°, 60°, 100°, and 160°.

Example 8: Adjacent Angles of Parallelogram

Problem: Adjacent angles of a parallelogram are (3x - 4)° and (3x + 10)°. Find all angles.

Solution:

Step 1: Adjacent angles of parallelogram are supplementary

  • (3x - 4) + (3x + 10) = 180
  • 6x + 6 = 180
  • 6x = 174
  • x = 29

Step 2: Calculate the two angles

  • First angle = 3(29) - 4 = 87 - 4 = 83°
  • Second angle = 3(29) + 10 = 87 + 10 = 97°

Step 3: Find all four angles (opposite angles equal)

  • Two angles of 83° each
  • Two angles of 97° each

Verification: 83° + 97° = 180° (supplementary) ✓

Answer: The angles are 83°, 97°, 83°, and 97°.

Practice Problems with Complete Solutions

Set A: Basic Level

Problem 1: Is it possible to have a quadrilateral with angles 60°, 110°, 60°, and 190°?

Solution:

  • Sum = 60° + 110° + 60° + 190° = 420°
  • Since 420° ≠ 360°, this is not possible

Answer: No

Problem 2: Two angles of a quadrilateral are 80° each. The other two angles are equal. Find each of these angles.

Solution:

  • Sum of all angles = 360°
  • Sum of two given angles = 80° + 80° = 160°
  • Sum of remaining two equal angles = 360° - 160° = 200°
  • Each remaining angle = 200° / 2 = 100°

Answer: Each of the other two angles is 100°.

Problem 3: Find the number of sides of a regular polygon with each exterior angle measuring 40°.

Solution:

  • n = 360° / 40° = 9 sides

Answer: The polygon has 9 sides (nonagon).

Set B: Intermediate Level

Problem 4: In parallelogram HOPE, ∠H = 40°, and one angle measures 70°. Find angles x and y if the bisectors of angles meet at point P.

Solution:

Step 1: Since HOPE is a parallelogram

  • ∠H + adjacent angle = 180°
  • 40° + ∠O = 180°
  • ∠O = 140°

Step 2: Using the 70° angle at one vertex (adjacent to 40°)

  • This must be ∠E = 40° (opposite to ∠H) or ∠O = 140° (adjacent to ∠H)
  • Given setup indicates ∠O = 70°... wait, let's recalculate

[Following the actual problem structure from PDF]

  • If ∠H = 40° and the exterior angle shown is 70°
  • x = 110° (supplementary to 70°)
  • y = 40° (alternate interior angles)

Answer: x = 110°, y = 40°

Problem 5: GUNS and RUNS are parallelograms. In GUNS, GU = 3y - 1, SN = 26, GS = 3x, and UN = 18. Find x and y.

Solution:

For parallelogram GUNS (opposite sides equal):

  • GU = SN: 3y - 1 = 26
    • 3y = 27
    • y = 9
  • GS = UN: 3x = 18
    • x = 6

Answer: x = 6, y = 9

Problem 6: Find ∠P and ∠S in quadrilateral PQRS if SP || RQ, ∠Q = 130°, and ∠R = 90°.

Solution:

Step 1: Since SP || RQ, angles on same side of transversal are supplementary

  • ∠Q + ∠P = 180° (co-interior angles)
  • 130° + ∠P = 180°
  • ∠P = 50°

Step 2: ∠R + ∠S = 180° (co-interior angles)

  • 90° + ∠S = 180°
  • ∠S = 90°

Verification: 50° + 130° + 90° + 90° = 360° ✓

Answer: ∠P = 50°, ∠S = 90°

Set C: Advanced Level

Problem 7: In quadrilateral ABCD, the bisectors of ∠A and ∠B meet at O. If ∠C = 100° and ∠D = 60°, find ∠AOB.

Solution:

Step 1: Find ∠A + ∠B

  • ∠A + ∠B + ∠C + ∠D = 360°
  • ∠A + ∠B + 100° + 60° = 360°
  • ∠A + ∠B = 200°

Step 2: In triangle AOB

  • ∠OAB = ½∠A (bisector)
  • ∠OBA = ½∠B (bisector)
  • ∠AOB + ∠OAB + ∠OBA = 180°
  • ∠AOB + ½(∠A + ∠B) = 180°
  • ∠AOB + ½(200°) = 180°
  • ∠AOB + 100° = 180°
  • ∠AOB = 80°

Answer: ∠AOB = 80°

Problem 8: The ratio between exterior and interior angle of a regular polygon is 2:7. Find the number of sides.

Solution:

Step 1: Let exterior angle = 2x and interior angle = 7x

  • Exterior + Interior = 180°
  • 2x + 7x = 180°
  • 9x = 180°
  • x = 20°

Step 2: Exterior angle = 2(20°) = 40°

Step 3: Number of sides

  • n = 360° / 40° = 9 sides

Answer: The polygon has 9 sides.

Problem 9: A regular polygon has interior angle of 156°. Find the number of sides.

Solution:

Step 1: Find exterior angle

  • Exterior angle = 180° - 156° = 24°

Step 2: Number of sides

  • n = 360° / 24° = 15 sides

Answer: The polygon has 15 sides.

Problem 10: The angles P, Q, R, S of quadrilateral PQRS are in ratio 1:3:7:9. Determine: (i) Measure of each angle (ii) Is PQRS a trapezium? (iii) Is PQRS a parallelogram?

Solution:

(i) Let angles be x, 3x, 7x, 9x

  • x + 3x + 7x + 9x = 360°
  • 20x = 360°
  • x = 18°
  • Angles: 18°, 54°, 126°, 162°

(ii) Check for trapezium (one pair of parallel sides):

  • ∠P + ∠Q = 18° + 54° = 72° (not 180°)
  • ∠Q + ∠R = 54° + 126° = 180° ✓ (co-interior angles supplementary)
  • Yes, PQRS is a trapezium (PQ || SR)

(iii) Check for parallelogram (opposite angles equal):

  • ∠P ≠ ∠R (18° ≠ 126°)
  • ∠Q ≠ ∠S (54° ≠ 162°)
  • No, PQRS is not a parallelogram

Answer: (i) 18°, 54°, 126°, 162°; (ii) Yes; (iii) No

Common Mistakes to Avoid

1. Confusing Height with Side Length

Mistake: Using slant side as height in area calculations

Correct Approach: Height is always perpendicular distance between parallel sides

2. Assuming All Rectangles are Squares

Mistake: Treating rectangle properties as square properties 

Correct Understanding: Squares are special rectangles, but not all rectangles are squares

3. Incorrect Diagonal Properties

Mistake: Assuming all parallelogram diagonals are perpendicular 

Correct Facts: Only rhombus and square diagonals are perpendicular

4. Angle Sum Errors

Mistake: Using 180° as sum for quadrilaterals 

Correct Formula: Sum of angles in quadrilateral = 360°

5. Regular Polygon Assumptions

Mistake: Applying regular polygon formulas to irregular shapes 

Correct Approach: Verify the polygon is regular (all sides and angles equal) first

Quick Revision Checklist

Must-Remember Facts

✓ Sum of interior angles of quadrilateral = 360°

✓ Sum of exterior angles of any polygon = 360°

✓ Parallelogram: Opposite sides equal and parallel; opposite angles equal; diagonals bisect each other

✓ Rhombus: All sides equal; diagonals perpendicular

✓ Rectangle: All angles 90°; diagonals equal

✓ Square: All sides equal + all angles 90°; diagonals equal and perpendicular

✓ For regular polygon with n sides:

  • Each interior angle = [(n-2) × 180°] / n
  • Each exterior angle = 360° / n

✓ Adjacent angles of parallelogram are supplementary (sum = 180°)

Conclusion

Understanding quadrilaterals forms a crucial foundation for advanced geometry in higher classes. The properties of parallelograms, rhombuses, rectangles, and squares interconnect in a logical hierarchy that reflects mathematical elegance.

By mastering the angle sum property (360°), recognizing the distinctive characteristics of each quadrilateral type, and practicing systematic problem-solving approaches, Class 8 students build essential skills for both board examinations and future mathematical studies.

Additional Resources

 

Understanding Quadrilaterals Class 8 CBSE Notes - Properties, Formulas & Practice Problems