Class 9 Mathematics Notes Chapter 8 (Chapter 8) – Examplar Problem (Englisha) Book

Alright class, let's dive into Chapter 8: Quadrilaterals, focusing on the key concepts from your NCERT Exemplar which are crucial for competitive government exams. Pay close attention, as understanding these properties and theorems is essential for problem-solving.

Chapter 8: Quadrilaterals - Detailed Notes for Exam Preparation

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.
• Elements: Sides (e.g., AB, BC, CD, DA), Angles (∠A, ∠B, ∠C, ∠D), Vertices (A, B, C, D), Diagonals (AC, BD).
• Adjacent Sides: Two sides with a common endpoint (e.g., AB and BC).
• Opposite Sides: Two sides with no common endpoint (e.g., AB and CD).
• Adjacent Angles: Two angles with a common side (e.g., ∠A and ∠B).
• Opposite Angles: Two angles with no common side (e.g., ∠A and ∠C).

2. Angle Sum Property of a Quadrilateral

• Theorem: The sum of the four interior angles of any quadrilateral is 360°.
  • Proof Idea: Draw one diagonal (e.g., AC). This divides the quadrilateral into two triangles (ΔABC and ΔADC). The sum of angles in each triangle is 180°. Adding the angles of both triangles gives the sum of angles of the quadrilateral.
  • ∠A + ∠B + ∠C + ∠D = 360°
• Application: Used to find unknown angles in a quadrilateral if others are known.

3. Types of Quadrilaterals

(a) Trapezium:

• Definition: A quadrilateral with exactly one pair of opposite sides parallel.
• Property: Sum of adjacent angles between the parallel sides is 180° (Consecutive Interior Angles). If AB || DC, then ∠A + ∠D = 180° and ∠B + ∠C = 180°.
• Isosceles Trapezium: A trapezium where the non-parallel sides are equal in length.
  • Properties: Base angles are equal (∠D = ∠C and ∠A = ∠B if AD = BC and AB || DC). Diagonals are equal (AC = BD).

(b) Parallelogram:

• Definition: A quadrilateral in which both pairs of opposite sides are parallel (AB || DC and AD || BC).
• Key Properties (Very Important):
  1. Opposite sides are equal (AB = DC, AD = BC).
  2. Opposite angles are equal (∠A = ∠C, ∠B = ∠D).
  3. Adjacent angles are supplementary (∠A + ∠B = 180°, ∠B + ∠C = 180°, etc.).
  4. Diagonals bisect each other (If diagonals AC and BD intersect at O, then AO = OC and BO = OD).
  5. Each diagonal divides the parallelogram into two congruent triangles (ΔABC ≅ ΔCDA, ΔABD ≅ ΔCDB).
• Conditions for a Quadrilateral to be a Parallelogram (Converse Theorems): A quadrilateral is a parallelogram if:
  1. Both pairs of opposite sides are equal.
  2. Both pairs of opposite angles are equal.
  3. The diagonals bisect each other.
  4. One pair of opposite sides is equal and parallel.

(c) Rectangle:

• Definition: A parallelogram in which one angle is 90°. (This implies all angles are 90°).
• Properties:
  1. All properties of a parallelogram apply.
  2. All angles are right angles (90°).
  3. Diagonals are equal in length (AC = BD).
  • Note: Diagonals bisect each other (property of parallelogram) and are equal.

(d) Rhombus:

• Definition: A parallelogram in which all sides are equal.
• Properties:
  1. All properties of a parallelogram apply.
  2. All sides are equal (AB = BC = CD = DA).
  3. Diagonals bisect each other at right angles (90°) (AC ⊥ BD).
  4. Diagonals bisect the angles at the vertices (e.g., diagonal AC bisects ∠A and ∠C).
  • Note: Diagonals bisect each other (property of parallelogram) and are perpendicular. They are generally not equal unless it's also a square.

(e) Square:

• Definition: A parallelogram with all sides equal and one angle 90°. (Alternatively, a rectangle with adjacent sides equal, or a rhombus with one angle 90°).
• Properties:
  1. All properties of a parallelogram apply.
  2. All properties of a rectangle apply.
  3. All properties of a rhombus apply.
  4. All sides are equal.
  5. All angles are 90°.
  6. Diagonals are equal.
  7. Diagonals bisect each other at right angles.
  8. Diagonals bisect the vertex angles (each bisected angle is 45°).

(f) Kite: (Less common in basic syllabus but appears in Exemplar)

• Definition: A quadrilateral in which two pairs of adjacent sides are equal, but opposite sides are unequal. (e.g., AB = AD and CB = CD).
• Properties:
  1. One diagonal is the perpendicular bisector of the other diagonal (AC ⊥ BD and AC bisects BD, if AB=AD and CB=CD).
  2. One diagonal bisects the angles at the vertices it joins (Diagonal AC bisects ∠A and ∠C).
  3. One pair of opposite angles (between unequal sides) are equal (∠B = ∠D).

4. The Mid-point Theorem

• 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 of it.
  • 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.
• Applications: Crucial for proving properties of quadrilaterals formed by joining mid-points, solving geometrical problems involving lengths and parallelism. For example, the quadrilateral formed by joining the mid-points of the sides of any quadrilateral is always a parallelogram.

5. Key Takeaways for Exams

• Memorize the definitions and distinct properties of each type of quadrilateral.
• Understand the hierarchy: Squares are rectangles and rhombuses; rectangles and rhombuses are parallelograms; parallelograms and isosceles trapeziums are trapeziums.
• Be able to apply the conditions required to prove a quadrilateral is a parallelogram, rectangle, rhombus, or square.
• The Mid-point Theorem and its converse are frequently tested, often in combination with properties of parallelograms.
• Practice problems involving finding angles, side lengths, and properties related to diagonals.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on Chapter 8 concepts, similar to what you might encounter:

1. The sum of all interior 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 in which diagonals are equal and bisect each other at right angles is a:
   (a) Rhombus
   (b) Square
   (c) Rectangle
   (d) Parallelogram

4. ABCD is a parallelogram. If ∠A = 70°, then ∠C is:
   (a) 70°
   (b) 110°
   (c) 90°
   (d) Cannot be determined

5. In a parallelogram ABCD, diagonals AC and BD intersect at O. If AO = 5 cm, then AC equals:
   (a) 5 cm
   (b) 10 cm
   (c) 2.5 cm
   (d) Cannot be determined

6. Which of the following is NOT necessarily a property of a rhombus?
   (a) All sides are equal.
   (b) Diagonals bisect each other at 90°.
   (c) Diagonals are equal.
   (d) Opposite angles are equal.

7. D and E are the mid-points of sides AB and AC of ΔABC respectively. If BC = 8 cm, then the length of DE is:
   (a) 8 cm
   (b) 16 cm
   (c) 4 cm
   (d) 2 cm

8. The quadrilateral formed by joining the mid-points of the sides of a rectangle, taken in order, is a:
   (a) Square
   (b) Rectangle
   (c) Rhombus
   (d) Parallelogram

9. In a trapezium ABCD with AB || DC, if ∠A = 100°, then ∠D equals:
   (a) 100°
   (b) 80°
   (c) 90°
   (d) 70°

10. If one pair of opposite sides of a quadrilateral are equal and parallel, the quadrilateral is a:
    (a) Trapezium
    (b) Parallelogram
    (c) Rhombus
    (d) Kite

Answer Key for MCQs:

1. (c) 360°
2. (d) 120° (360° - 75° - 90° - 75° = 120°)
3. (b) Square (Equal diagonals => Rectangle; Diagonals bisect at 90° => Rhombus. Both conditions => Square)
4. (a) 70° (Opposite angles of a parallelogram are equal)
5. (b) 10 cm (Diagonals of a parallelogram bisect each other, so AC = 2 * AO)
6. (c) Diagonals are equal (This is true for a square or rectangle, but not necessarily a rhombus)
7. (c) 4 cm (Mid-point theorem: DE = ½ BC)
8. (c) Rhombus (Joining mid-points of a rectangle gives a rhombus)
9. (b) 80° (Adjacent angles between parallel sides are supplementary: ∠A + ∠D = 180°)
10. (b) Parallelogram (This is one of the conditions for a quadrilateral to be a parallelogram)

Make sure you thoroughly understand these concepts and practice problems from your Exemplar book. Good luck with your preparation!