Class 8 Maths Quadrilaterals Notes

Class 8 Maths Quadrilaterals Notes

Chapter 4: Quadrilaterals

Overview:
This chapter introduces quadrilaterals, a group of four-sided polygons, and explains their properties, types, and angle relationships. Students learn to identify, classify, and solve problems involving quadrilaterals using geometrical rules and theorems.

Key Concepts

  1. Definition:
    • A quadrilateral is a polygon with 4 sides, 4 vertices, and 4 angles.
  2. Types of Quadrilaterals: TypePropertiesParallelogramOpposite sides are equal and parallel; opposite angles equal; diagonals bisect each otherRectangleAll properties of parallelogram + all angles 90°; diagonals equalRhombusAll sides equal + opposite sides parallel; diagonals perpendicular bisectorsSquareAll properties of rectangle and rhombus; all sides equal, all angles 90°, diagonals equal and perpendicularTrapeziumOnly one pair of opposite sides parallelKiteTwo pairs of adjacent sides equal; diagonals perpendicular; one diagonal bisected
  3. Sum of Interior Angles:
    • Formula: (n−2)×180∘(n-2) \times 180^\circ(n−2)×180∘, where n=4n = 4n=4
    • For quadrilateral: (4−2)×180=360∘(4-2) \times 180 = 360^\circ(4−2)×180=360∘
  4. Sum of Exterior Angles:
    • Always 360°, irrespective of quadrilateral type
  5. Diagonals of Quadrilaterals:
    • Parallelogram: bisect each other
    • Rectangle: equal and bisect each other
    • Rhombus: perpendicular bisectors
    • Square: equal, perpendicular, bisect each other
  6. Special Properties:
    • Opposite sides of parallelogram are equal
    • Adjacent angles of a parallelogram sum to 180°
    • In a rectangle, the diagonals are equal
    • In a rhombus, diagonals divide the angles into two equal parts

Important Points to Remember:

  • Sum of interior angles = 360°
  • Quadrilaterals can be classified based on sides and angles
  • Diagonals give useful information for solving geometry problems
  • Knowing properties helps identify unknown angles or sides in problems

Examples:

  1. Sum of Interior Angles:
    • A quadrilateral has 4 sides → Sum = (4−2)×180=360°(4-2) \times 180 = 360°(4−2)×180=360°
  2. Finding an Unknown Angle:
    • In a parallelogram, one angle is 70° → opposite angle = 70°
    • Adjacent angles = 180° − 70° = 110°
  3. Using Diagonal Properties:
    • In a rhombus, if one diagonal = 8 cm and other = 6 cm, they bisect each other at 90° → halves = 4 cm and 3 cm each

Questions

A. Very Short Answer Questions (1–10)

  1. Define a quadrilateral.
  2. How many sides does a quadrilateral have?
  3. What is the sum of interior angles of a quadrilateral?
  4. What is the sum of exterior angles of a quadrilateral?
  5. Name the types of quadrilaterals.
  6. In a parallelogram, are opposite sides equal?
  7. In a rectangle, are diagonals equal?
  8. In a rhombus, are diagonals perpendicular?
  9. How many right angles are there in a square?
  10. Is a square a rectangle, a rhombus, or both?

B. Short Answer Questions (11–25)

  1. A quadrilateral has three angles of 90°, 80°, and 110°. Find the fourth angle.
  2. In a parallelogram, one angle is 70°. Find all other angles.
  3. Name a quadrilateral with only one pair of opposite sides parallel.
  4. Name a quadrilateral whose diagonals bisect each other and are equal.
  5. Name a quadrilateral whose diagonals bisect each other and are perpendicular.
  6. A rectangle has length 8 cm and width 5 cm. Find the perimeter.
  7. Find the area of a rectangle with length 10 cm and width 6 cm.
  8. In a rhombus, if diagonals are 8 cm and 6 cm, find the length of each side.
  9. Check whether all squares are rectangles.
  10. Check whether all rectangles are squares.
  11. Find the perimeter of a square with side 7 cm.
  12. Find one pair of opposite angles in a parallelogram if one angle is 120°.
  13. A quadrilateral has interior angles 90°, 85°, 95°, and x°. Find x.
  14. Find the sum of two adjacent angles of a parallelogram if one angle is 60°.
  15. A kite has diagonals 10 cm and 8 cm. Are they perpendicular?

C. Word Problems / Application Questions (26–40)

  1. The angles of a quadrilateral are in the ratio 2:3:4:5. Find all angles.
  2. The opposite sides of a parallelogram are 12 cm and 8 cm. Find its perimeter.
  3. The diagonals of a rhombus are 12 cm and 16 cm. Find its area.
  4. In a rectangle, the length = 15 cm, width = 10 cm. Find the diagonal.
  5. A square has a side of 9 cm. Find the length of its diagonal.
  6. In a parallelogram, one angle = 110°. Find the other angles.
  7. Find the perimeter of a rhombus with side 10 cm.
  8. A trapezium has parallel sides 8 cm and 5 cm, height 4 cm. Find area.
  9. The diagonals of a rectangle are 10 cm long. If the length = 6 cm, find width.
  10. A square and a rhombus have the same perimeter 20 cm. Find side length.
  11. The sum of two adjacent angles of a parallelogram is 180°. If one angle is x°, find the other.
  12. A kite has two pairs of equal adjacent sides, 5 cm and 8 cm. Find perimeter.
  13. A quadrilateral has interior angles 85°, 95°, 100°, and x°. Find x.
  14. In a rhombus, one angle = 60°. Find the other angles.
  15. A parallelogram has base 12 cm and height 5 cm. Find its area.

D. Higher Order / Thinking Questions (41–50)

  1. Name a quadrilateral whose diagonals are neither equal nor perpendicular.
  2. Find the diagonal of a square with side 12 cm.
  3. In a rhombus, diagonals are 10 cm and 24 cm. Find each side.
  4. In a rectangle, length = 8 cm, width = 6 cm. Find perimeter and area.
  5. Show that in a parallelogram, opposite angles are equal.
  6. A quadrilateral has two right angles and two obtuse angles. Name it.
  7. The angles of a quadrilateral are in the ratio 1:2:3:4. Find the largest angle.
  8. The sides of a rectangle are in the ratio 3:2. If perimeter = 20 cm, find length and width.
  9. Find the perimeter of a parallelogram whose sides are 8 cm and 6 cm.
  10. Diagonals of a rhombus divide it into four right triangles. Explain why.

Answers

A. Very Short Answer Questions (1–10)

  1. Quadrilateral: A polygon with 4 sides, 4 vertices, and 4 angles.
  2. A quadrilateral has 4 sides.
  3. Sum of interior angles = (4−2)×180°=360°(4-2) × 180° = 360°(4−2)×180°=360°
  4. Sum of exterior angles = 360°
  5. Types: Parallelogram, Rectangle, Rhombus, Square, Trapezium, Kite
  6. In a parallelogram, opposite sides are equal ✅
  7. In a rectangle, diagonals are equal ✅
  8. In a rhombus, diagonals are perpendicular ✅
  9. A square has 4 right angles (90° each)
  10. A square is both a rectangle and a rhombus ✅

B. Short Answer Questions (11–25)

  1. Sum of angles = 360° → 90 + 80 + 110 + x = 360 → x = 360−280 = 80°
  2. Parallelogram opposite angles equal: 70°, 110°, 70°, 110°
  3. Quadrilateral with one pair of parallel sides: Trapezium
  4. Quadrilateral whose diagonals bisect each other and are equal: Rectangle
  5. Quadrilateral whose diagonals bisect each other and are perpendicular: Rhombus
  6. Rectangle perimeter = 2(l + w) = 2(8 + 5) = 26 cm
  7. Rectangle area = l × w = 10 × 6 = 60 cm²
  8. Rhombus side = √[(d1/2)² + (d2/2)²] = √(4² + 3²) = √(16+9)=√25 = 5 cm
  9. Yes, all squares are rectangles (all angles 90°, opposite sides equal)
  10. No, not all rectangles are squares (sides may not be equal)
  11. Square perimeter = 4 × side = 4 × 7 = 28 cm
  12. One angle = 120° → opposite = 120°, adjacent = 60° → angles = 120°,60°,120°,60°
  13. 90 + 85 + 95 + x = 360 → x = 360−270 = 90°
  14. Adjacent angles sum = 180 → 180 − 60 = 120°
  15. Kite diagonals are perpendicular → Yes

C. Word Problems / Applications (26–40)

  1. Ratio 2:3:4:5 → sum = 2x+3x+4x+5x=14x=360 → x=25 → angles = 50°,75°,100°,125°
  2. Parallelogram perimeter = 2(a+b)=2(12+8)=40 cm
  3. Rhombus area = ½ × d1 × d2 = ½ × 12 × 16 = 96 cm²
  4. Rectangle diagonal = √(l² + w²) = √(15²+10²)=√(225+100)=√325 ≈ 18.03 cm
  5. Square diagonal = √2 × side = √2 × 9 ≈ 12.73 cm
  6. Parallelogram one angle = 110° → opposite = 110°, adjacent = 70° → angles = 110°,70°,110°,70°
  7. Rhombus perimeter = 4 × side = 4 × 10 = 40 cm
  8. Trapezium area = ½ × (sum of parallel sides) × height = ½ × (8+5) × 4 = 26 cm²
  9. Rectangle diagonals = √(l² + w²) = 10 → 10² = 64 + w² → 100−64=w² → w²=36 → w=6 cm
  10. Square perimeter = 20 → side = 20 ÷ 4 = 5 cm → rhombus side = 5 cm
  11. Adjacent angles sum = 180 → if one angle x°, other = 180−x°
  12. Kite perimeter = 2(5+8)=2×13=26 cm
  13. Quadrilateral angles sum = 360 → 85+95+100+x=360 → x=360−280=80°
  14. Rhombus one angle = 60° → opposite = 60°, other two = 180−60=120° each → angles = 60°,120°,60°,120°
  15. Parallelogram area = base × height = 12 × 5 = 60 cm²

D. Higher Order / Thinking Questions (41–50)

  1. Quadrilateral whose diagonals are neither equal nor perpendicular: Trapezium (non-isosceles)
  2. Square diagonal = √2 × side = √2 × 12 ≈ 16.97 cm
  3. Rhombus side = √[(d1/2)² + (d2/2)²] = √(5² + 12²)=√(25+144)=√169=13 cm
  4. Rectangle perimeter = 2(l+w)=2(8+6)=28 cm; area = 8×6=48 cm²
  5. In parallelogram, opposite angles equal → proved by geometry: ∠A=∠C, ∠B=∠D
  6. Quadrilateral with 2 right + 2 obtuse angles = Trapezium
  7. Angles ratio 1:2:3:4 → sum=10x=360 → x=36 → largest = 4x=144°
  8. Rectangle sides ratio 3:2, perimeter=20 → 2(3x+2x)=20 → 10x=20 → x=2 → length=6 cm, width=4 cm
  9. Parallelogram perimeter = 2(a+b)=2(8+6)=28 cm
  10. Diagonals of rhombus divide it into 4 right triangles → because diagonals are perpendicular bisectors → each triangle has right angle at intersection