Class 9 Mathematics Notes Chapter 8 (Quadrilaterals) – Mathematics Book

Alright class, let's focus on Chapter 8: Quadrilaterals. This is a fundamental chapter in geometry, and understanding its concepts thoroughly is crucial not just for your Class 9 exams but also forms a base for many questions in competitive government exams.

We'll break down the key ideas, properties, and theorems. Pay close attention!

Chapter 8: Quadrilaterals - Detailed Notes

1. Introduction

• Quadrilateral: A closed figure formed by joining four points (no three of which are collinear) in order. It has four sides, four angles, and four vertices.
• Angle Sum Property: The sum of the four interior angles of any quadrilateral is always 360°.
  • Proof Idea: Draw a diagonal. This divides the quadrilateral into two triangles. The sum of angles in each triangle is 180°. Adding them gives 180° + 180° = 360°.

2. Types of Quadrilaterals

We classify quadrilaterals based on their sides and angles:

• a) Trapezium: A quadrilateral with at least one pair of opposite sides parallel.

  • Isosceles Trapezium: A trapezium where the non-parallel sides are equal. Its base angles are equal, and diagonals are equal. (Note: NCERT focuses mainly on the basic definition).

• b) Parallelogram: A quadrilateral where both pairs of opposite sides are parallel.

  • This is a very important type, with many properties and theorems associated with it.

• c) Rectangle: A parallelogram with one angle equal to 90°. (This implies all angles are 90°).

• d) Rhombus: A parallelogram with all four sides equal.

• e) Square: A parallelogram with all four sides equal AND one angle equal to 90°.

  • Alternatively, a square is a rectangle with adjacent sides equal, OR a rhombus with one angle 90°. It inherits properties from both rectangles and rhombuses.

• f) Kite: A quadrilateral with two distinct pairs of equal adjacent sides. (AB=AD and BC=CD, but AB≠BC).

  • Properties: Diagonals are perpendicular; one diagonal bisects the other; one diagonal bisects the angles at the vertices it joins; one pair of opposite angles (between unequal sides) are equal.

3. Properties of a Parallelogram

These are extremely important and frequently tested:

• Theorem 8.1: A diagonal of a parallelogram divides it into two congruent triangles. (Proof uses ASA or SSS congruence).
• Theorem 8.2: In a parallelogram, opposite sides are equal. (Consequence of Theorem 8.1).
• Theorem 8.3 (Converse of 8.2): If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.
• Theorem 8.4: In a parallelogram, opposite angles are equal.
• Theorem 8.5 (Converse of 8.4): If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.
• Theorem 8.6: The diagonals of a parallelogram bisect each other. (Meaning they cut each other into two equal halves at the point of intersection).
• Theorem 8.7 (Converse of 8.6): If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
• Theorem 8.8: A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel. (This is a very useful condition for proofs).
• Consecutive Angles: Consecutive angles (angles next to each other) in a parallelogram are supplementary (add up to 180°). This follows from the property of parallel lines intersected by a transversal.

4. Properties of Special Parallelograms

• Rectangle:

  • All properties of a parallelogram apply.
  • All angles are 90°.
  • Diagonals are equal.
  • Diagonals bisect each other (from parallelogram property).

• Rhombus:

  • All properties of a parallelogram apply.
  • All sides are equal.
  • Diagonals bisect each other at right angles (90°).
  • Diagonals bisect the angles at the vertices.

• Square:

  • All properties of a parallelogram, rectangle, and rhombus apply.
  • All sides are equal.
  • All angles are 90°.
  • Diagonals are equal.
  • Diagonals bisect each other at right angles (90°).
  • Diagonals bisect the angles at the vertices (each bisected angle is 45°).

5. The Mid-point Theorem

This theorem and its converse are crucial for many geometry problems.

• Theorem 8.9 (Mid-point Theorem): The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and is half the length of the third side.

  • If D and E are mid-points of sides AB and AC of ΔABC respectively, then DE || BC and DE = ½ BC.

• Theorem 8.10 (Converse of Mid-point Theorem): The line drawn through the mid-point of one side of a triangle, parallel to another side, bisects the third side.

  • If D is the mid-point of AB in ΔABC, and a line through D parallel to BC intersects AC at E, then E is the mid-point of AC.

Summary for Exam Preparation:

• Know the definitions and basic properties (sides, angles) of all quadrilateral types.
• Memorize the properties of a parallelogram and the conditions required to prove a quadrilateral is a parallelogram (Theorems 8.1 to 8.8).
• Understand how Rectangle, Rhombus, and Square are special cases of parallelograms and their additional unique properties (especially regarding diagonals).
• Master the Mid-point Theorem and its Converse – be ready to apply them in problems involving triangles and quadrilaterals formed by joining mid-points.
• Practice proofs and numerical problems based on these theorems and properties.

Multiple Choice Questions (MCQs)

Here are 10 MCQs to test your understanding. These are typical of the foundational questions you might encounter.

1. The sum of angles of a quadrilateral is:
   (a) 180°
   (b) 270°
   (c) 360°
   (d) 540°

2. Three angles of a quadrilateral are 75°, 90°, and 75°. The fourth angle is:
   (a) 90°
   (b) 95°
   (c) 105°
   (d) 120°

3. A quadrilateral with only one pair of opposite sides parallel is called a:
   (a) Parallelogram
   (b) Trapezium
   (c) Rhombus
   (d) Kite

4. Which of the following is NOT necessarily a property of a parallelogram?
   (a) Opposite sides are equal
   (b) Diagonals bisect each other
   (c) Opposite angles are equal
   (d) Diagonals are equal

5. The diagonals of a quadrilateral bisect each other at right angles. This quadrilateral is a:
   (a) Rectangle
   (b) Parallelogram
   (c) Rhombus
   (d) Trapezium

6. In triangle ABC, D and E are mid-points of sides AB and AC respectively. If BC = 8 cm, then the length of DE is:
   (a) 8 cm
   (b) 4 cm
   (c) 6 cm
   (d) 2 cm

7. A square has the properties of:
   (a) A parallelogram only
   (b) A rhombus and a rectangle
   (c) A trapezium and a kite
   (d) A rectangle only

8. If one angle of a parallelogram is 60°, what is the measure of its adjacent angle?
   (a) 60°
   (b) 90°
   (c) 120°
   (d) 30°

9. In a quadrilateral ABCD, AB = CD and AD = BC. What type of quadrilateral is ABCD?
   (a) Trapezium
   (b) Parallelogram
   (c) Kite
   (d) Cannot be determined

10. A line is drawn through the mid-point of one side of a triangle, parallel to the second side. What does it do to the third side?
    (a) Trisects it
    (b) Is perpendicular to it
    (c) Bisects it
    (d) Is equal to half of it

Answer Key for MCQs:

1. (c) 360°
2. (d) 120° (360° - 75° - 90° - 75° = 120°)
3. (b) Trapezium
4. (d) Diagonals are equal (This is true for rectangles and squares, but not all parallelograms)
5. (c) Rhombus (If diagonals were also equal, it would be a square)
6. (b) 4 cm (By Mid-point Theorem, DE = ½ BC)
7. (b) A rhombus and a rectangle
8. (c) 120° (Adjacent angles in a parallelogram are supplementary: 180° - 60° = 120°)
9. (b) Parallelogram (If both pairs of opposite sides are equal, it's a parallelogram - Theorem 8.3)
10. (c) Bisects it (Converse of Mid-point Theorem)

Make sure you understand the reasoning behind each answer. Revise these notes thoroughly. Good luck with your preparation!