NCERT Notes for Class 9 Maths Chapter 9 Quadrilaterals
Class 9 Maths Chapter 9 Quadrilaterals Notes
Chapter Name  Quadrilaterals Notes 
Class  CBSE Class 9 
Textbook Name  Mathematics Class 9 
Related Readings 

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.
g., AB = DC and AD = BC  Consecutive angles are supplementary.
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.
Midpoint Theorem
Theorem 8.9: The line segment joining the midpoints of two sides of a triangle is parallel to the third.
Given: A triangle ABC, E and F are midpoints of sides AB and AC respectively.
i.e., AE = EB and AF = FC
To Prove:
(i) EF  BC
(ii) EF = 12 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 midpoint of AC)
∠AFE = ∠CFD (Vertically opposite angles)
∠EAF = ∠DCF (Alternate interior angles)
∴ Î”AEF = Î”CDF (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 = 1/2 BC
Converse of MidPoint Theorem
Theorem 8.10: The line drawn through the midpoint of one side of a triangle, parallel to another side bisects the third side. ‘
Given: Î”ABC 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 Î”AEF and Î”CDF, ∠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)