Class 8 Notes

Class 8 Mathematics Notes Chapter 3 (Understanding Quadrilaterals) – Mathematics Book

Philoid

Philoid

— 5 min read

Mathematics
Alright class, let's focus on Chapter 3, 'Understanding Quadrilaterals'. This is a foundational chapter in geometry, and understanding these concepts well is crucial not just for your Class 8 exams but also for various government exams where geometry questions frequently appear. We'll break it down systematically.

Chapter 3: Understanding Quadrilaterals - Detailed Notes

1. Introduction to Polygons

  • Curve: Any drawing done without lifting the pencil.
  • Simple Curve: A curve that does not cross itself.
  • Closed Curve: A curve that begins and ends at the same point.
  • Simple Closed Curve: A closed curve that does not cross itself.
  • Polygon: A simple closed curve made up entirely of line segments.
    • The line segments are called sides.
    • The point where two sides meet is called a vertex (plural: vertices).
    • Any two sides with a common endpoint are adjacent sides.
    • The endpoints of the same side are adjacent vertices.

2. Classification of Polygons

Polygons are classified based on the number of sides (or vertices):

Number of Sides Name
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
n n-gon

3. Diagonals

  • A diagonal is a line segment connecting two non-consecutive vertices of a polygon.
  • Example: In quadrilateral ABCD, AC and BD are diagonals. AB, BC, CD, DA are sides (connecting consecutive vertices).
  • A triangle has no diagonals.
  • A quadrilateral has 2 diagonals.
  • A pentagon has 5 diagonals.
  • (Formula for number of diagonals in an n-sided polygon: n(n-3)/2 - useful for competitive exams)

4. Convex and Concave Polygons

  • Convex Polygon: A polygon in which all interior angles are less than 180°. Equivalently, all diagonals lie entirely inside the polygon. Most polygons we study are convex.
  • Concave Polygon: A polygon in which at least one interior angle is greater than 180° (a reflex angle). Equivalently, at least one diagonal lies partially or wholly outside the polygon.

5. Regular and Irregular Polygons

  • Regular Polygon: A polygon that is both:
    • Equilateral: All sides are equal in length.
    • Equiangular: All interior angles are equal in measure.
    • Examples: Equilateral triangle, Square, Regular Pentagon, Regular Hexagon.
  • Irregular Polygon: A polygon that is not regular (either not equilateral, or not equiangular, or neither).
    • Examples: Scalene triangle, Rectangle (equiangular but not equilateral), Rhombus (equilateral but not equiangular), most general quadrilaterals.

6. Angle Sum Property of Polygons

  • Triangle: The sum of the measures of the three interior angles of a triangle is always 180°.
  • Quadrilateral: The sum of the measures of the four interior angles of a quadrilateral is always 360°.
    • Derivation: Draw one diagonal. It divides the quadrilateral into two triangles. Sum of angles = 180° + 180° = 360°.
  • General Polygon (n-sided): The sum of the measures of the interior angles of a convex polygon with 'n' sides is given by the formula:
    Sum of Interior Angles = (n - 2) × 180°
    • Derivation: Pick one vertex. Draw all possible diagonals from that vertex. This divides the n-sided polygon into (n-2) triangles. The sum of angles is (n-2) times the angle sum of a triangle.

7. Sum of Exterior Angles of a Polygon

  • An exterior angle is formed by extending one side of the polygon and the adjacent side.
  • The sum of the measures of the exterior angles (one at each vertex, taken in order) of any convex polygon is always 360°, regardless of the number of sides.
  • For a regular n-sided polygon:
    • Measure of each exterior angle = 360° / n
    • Measure of each interior angle = 180° - (Exterior Angle) OR [(n-2) × 180°] / n

8. Kinds of Quadrilaterals

Based on the nature of sides or angles, quadrilaterals can be classified:

  • Trapezium: A quadrilateral with at least one pair of parallel sides.
    • Isosceles Trapezium: A trapezium where the non-parallel sides are equal in length. Base angles are also equal.
  • Kite: A quadrilateral with exactly two distinct pairs of equal adjacent sides.
    • Properties:
      • Diagonals are perpendicular to each other.
      • One of the diagonals bisects the other diagonal.
      • One of the diagonals bisects the angles at the vertices it joins. (The diagonal connecting the vertices where unequal sides meet).
      • One pair of opposite angles (between unequal sides) are equal.
  • Parallelogram: A quadrilateral whose opposite sides are parallel.
    • Properties:
      1. Opposite sides are equal in length.
      2. Opposite angles are equal in measure.
      3. Adjacent angles are supplementary (their sum is 180°).
      4. Diagonals bisect each other (they cut each other into two equal parts at their intersection point).
    • Conditions for a quadrilateral to be a parallelogram (converse properties): If any one of these is true, it's a parallelogram:
      • Each pair of opposite sides is equal.
      • Each pair of opposite angles is equal.
      • Diagonals bisect each other.
      • One pair of opposite sides is equal AND parallel.

9. Special Parallelograms

These are parallelograms with additional properties:

  • Rhombus: A parallelogram with all four sides equal.
    • Inherited Properties: All properties of a parallelogram apply.
    • Additional Properties:
      1. Diagonals bisect each other at right angles (90°).
      2. Diagonals bisect the angles at the vertices.
  • Rectangle: A parallelogram with one angle (and hence all angles) equal to 90°.
    • Inherited Properties: All properties of a parallelogram apply.
    • Additional Properties:
      1. All angles are right angles (90°).
      2. Diagonals are equal in length.
  • Square: A parallelogram that is both a rhombus and a rectangle. It has all the properties of parallelograms, rhombuses, and rectangles.
    • Properties:
      1. All sides are equal.
      2. All angles are 90°.
      3. Diagonals bisect each other.
      4. Diagonals are equal.
      5. Diagonals bisect each other at right angles (90°).
      6. Diagonals bisect the angles at the vertices (each bisected angle is 45°).

Hierarchy/Relationships:

  • Squares are special types of Rectangles.
  • Squares are special types of Rhombuses.
  • Rectangles and Rhombuses are special types of Parallelograms.
  • Parallelograms are special types of Trapeziums (using the "at least one pair" definition).
  • Kites are generally not Parallelograms (unless they are also Rhombuses).

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on this chapter for your practice:

  1. What is the sum of the interior angles of a convex polygon with 7 sides (Heptagon)?
    A) 360°
    B) 720°
    C) 900°
    D) 1080°

  2. How many diagonals does a regular hexagon have?
    A) 6
    B) 9
    C) 12
    D) 5

  3. Each exterior angle of a regular polygon is 40°. How many sides does the polygon have?
    A) 6
    B) 8
    C) 9
    D) 10

  4. In a parallelogram PQRS, if ∠P = 80°, what is the measure of ∠Q?
    A) 80°
    B) 100°
    C) 90°
    D) 180°

  5. Which of the following quadrilaterals has diagonals that are perpendicular bisectors of each other?
    A) Rectangle
    B) Trapezium
    C) Kite
    D) Rhombus

  6. A quadrilateral has three acute angles each measuring 75°. What is the measure of the fourth angle?
    A) 105°
    B) 135°
    C) 150°
    D) 75°

  7. Which statement is ALWAYS true?
    A) Every rhombus is a square.
    B) Every parallelogram is a rectangle.
    C) Every square is a parallelogram.
    D) Every trapezium is a parallelogram.

  8. The adjacent angles of a parallelogram are in the ratio 2:3. What is the measure of the smaller angle?
    A) 72°
    B) 108°
    C) 60°
    D) 90°

  9. A quadrilateral in which exactly two distinct pairs of adjacent sides are equal is called a:
    A) Parallelogram
    B) Rectangle
    C) Kite
    D) Trapezium

  10. In rectangle ABCD, the diagonals AC and BD intersect at O. If AC = 10 cm, what is the length of BO?
    A) 10 cm
    B) 5 cm
    C) 20 cm
    D) Cannot be determined

Answers to MCQs:

  1. C ( (7-2) * 180° = 5 * 180° = 900° )
  2. B ( n(n-3)/2 = 6(6-3)/2 = 6*3/2 = 9 )
  3. C ( Sides n = 360° / Exterior Angle = 360° / 40° = 9 )
  4. B ( Adjacent angles in a parallelogram are supplementary: 180° - 80° = 100° )
  5. D ( Rhombus has diagonals that are perpendicular bisectors. Kite has perpendicular diagonals, but only one bisects the other. Square also fits, but Rhombus is the broader category listed here with this specific property distinguishing it from a general parallelogram or rectangle).
  6. B ( Sum of angles = 360°. Sum of three angles = 3 * 75° = 225°. Fourth angle = 360° - 225° = 135° )
  7. C ( A square fits the definition of a parallelogram - opposite sides parallel).
  8. A ( Let angles be 2x and 3x. Adjacent angles are supplementary: 2x + 3x = 180° => 5x = 180° => x = 36°. Smaller angle = 2x = 2 * 36° = 72° )
  9. C ( Definition of a Kite )
  10. B ( Diagonals of a rectangle are equal and bisect each other. So, BD = AC = 10 cm. BO is half of BD, so BO = 10/2 = 5 cm )

Study these notes carefully, focusing on the properties and definitions. Practice drawing the figures and identifying their characteristics. Good luck with your preparation!

Read more