## CBSE Class 9 Maths Chapter 9 Notes Quadrilaterals

### Quadrilaterals Class 9 Notes Understanding the Lesson

Quadrilateral

A plane figure bounded by four line segments is called quadrilateral.

Properties:

- It has four sides.
- It has four vertices or comers.
- It has two diagonals.
- The sum of four interior angles is equal to 360°.

In quadrilateral ABCD, AB, BC, CD and DA are sides; AC and BD are diagonals and

∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°.

Types of Quadrilaterals

1. Parallelogram: A quadrilateral whose each pair of opposite sides are parallel.

- AB || DC
- AD || BC

2. Rectangle: A parallelogram whose one angle is 90°. Diagonals are equal.

3. Rhombus: A parallelogram whose adjacent sides are equal.

Note: Diagonal bisect each other at 90°.

4. Square: A rectangle whose adjacent sides are equal (four sides are equal). Diagonal bisect each other at 90°.

5. Trapezium: A quadrilateral whose one pair of opposite sides are parallel. AB || DC

6. Kite: It has two pair of adjacent sides that are equal in length but opposite sides are unequal.

Note:

- One of the diagonal bisects the other at right angle.
- One pair of opposite angles are equal.

Properties of a Parallelogram

- Opposite sides are equal.

e.g., AB = DC and AD = BC - Consecutive angles are supplementary.

e.g., ∠A + ∠D = 180° - Diagonals of parallelogram bisect each other.
- Diagonal divide it into two congruent triangles. A B

Theorem 8.1: A diagonal of a parallelogram divides it into two congruent triangles.

Theorem 8,2: In a parallelogram, opposite sides are equal.

Theorem 8.3: 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: If in a quadrilateral, each pair of opposite angles of a quadrilateral is equal then it is a parallelogram.

Theorem 8.6: The diagonals of a parallelogram bisect each other.

Theorem 8.7: If the diagonals of 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.

Mid-point Theorem

Theorem 8.9: The line segment joining the mid-points of two sides of a triangle is parallel to the third.

Given: A triangle ABC, E and F are mid-points of sides AB and AC respectively.

i.e., AE = EB and AF = FC

To Prove:

(i) EF || BC

(ii) EF = \(\frac{1}{2}\) BC

Construction: Draw a line through C parallel to AB and extend EF which intersect at D.

Proof: (i) In AAEF and ACDF,

AF = CF (F is the mid-point of AC)

∠AFE = ∠CFD (Vertically opposite angles)

∠EAF = ∠DCF (Alternate interior angles)

∴ ΔAEF = ΔCDF (by ASA congruency)

∴ AE = CD (by CPCT)

and BE = CD (AE = BE)

EF = FD (by CPCT);

Hence, BCDE is a parallelogram.

ED || BC )

∴ EF || BC

(ii) BCDE is a parallelogram.

DE = BC

EF + FD = BC

2EF = BC

EF=\(\frac{1}{2}\)BC

Converse of Mid-Point Theorem

Theorem 8.10: The line drawn through the mid-point of one side of a triangle, parallel to another side bisects the third side. ‘

Given: ΔABC in which E is the mid point of AB.

EF || BC

To Prove: AF = FC

Construction: Draw CD || AB and extend EF which intersect at D.

Proof: EF || BC

∴ ED || BC

AB || CD

⇒ BE || CD

∴ BCDE is a parallelogram.

Now in ΔAEF and ΔCDF, ∠AFE = ∠CFD (Vertically opposite angles)

∠EAF = ∠DCF (Alternate interior angles)

AE = CD (BE = AE opposite side of a parallelogram and BE = CD

∴ AAEF ≅ ACDF (by AAS congruency)

Hence AF = FC (by CPCT)