• NEET
      • Class 11th
      • Class 12th
      • Class 12th Plus
    • JEE
      • Class 11th
      • Class 12th
      • Class 12th Plus
    • Class 6-10
      • Class 6th
      • Class 7th
      • Class 8th
      • Class 9th
      • Class 10th
    • View All Options
      • Online Courses
      • Offline Courses
      • Distance Learning
      • Hindi Medium Courses
    • NEET
      • Class 11th
      • Class 12th
      • Class 12th Plus
    • JEE (Main+Advanced)
      • Class 11th
      • Class 12th
      • Class 12th Plus
    • JEE Main
      • Class 11th
      • Class 12th
      • Class 12th Plus
    • CUET
      • Class 12th
  • NEW
    • JEE MAIN 2025
    • NEET
      • 2024
      • 2023
      • 2022
    • Class 6-10
    • JEE Main
      • Previous Year Papers
      • Sample Papers
      • Result
      • Analysis
      • Syllabus
      • Exam Date
    • JEE Advanced
      • Previous Year Papers
      • Sample Papers
      • Mock Test
      • Result
      • Analysis
      • Syllabus
      • Exam Date
    • NEET
      • Previous Year Papers
      • Sample Papers
      • Mock Test
      • Result
      • Analysis
      • Syllabus
      • Exam Date
    • NCERT Solutions
      • Class 6
      • Class 7
      • Class 8
      • Class 9
      • Class 10
      • Class 11
      • Class 12
    • CBSE
      • Notes
      • Sample Papers
      • Question Papers
    • Olympiad
      • NSO
      • IMO
      • NMTC
    • ALLEN e-Store
    • AOSAT
    • ALLEN for Schools
    • About ALLEN
    • Blogs
    • News
    • Careers
    • Request a call back
    • Book home demo
CBSE Notes
Class 9
Maths
Chapter 8 Quadrilaterals

CBSE Notes For Class 9 Maths Chapter 8 Quadrilaterals 

1.0Introduction to Quadrilaterals 

The word "Quadrilateral" originates from two Latin words: "quadri", meaning four, and "latus" referring to the side. This makes it a 4-sided figure in geometry. These can be of different shapes and sizes: 

Quadrilaterals

2.0Download CBSE Notes for Class 9 Maths Chapter 8 Quadrilaterals - Free PDF

Students can now download free, well-structured PDF notes for CBSE Class 9 Maths Chapter 8 Quadrilaterals, designed to help simplify and strengthen their understanding of this important geometry chapter.

Class 9 Maths Chapter 8 Revision Notes:

3.0CBSE Class 9 Maths Chapter 8 Quadrilaterals - Revision Notes

What is a Parallelogram? 

A parallelogram is a type of quadrilaterals which have opposite sides equal & parallel to each other. Some properties of parallelograms are as follows: 

Properties of A Parallelogram

Theorem-1: A diagonal of a parallelogram divides it into congruent triangles. 

To prove: ABC ≅ CDA 

Parallelogram

Given: ABCD is a parallelogram, which means CD is parallel to AB

Proof: In ABC and CDA

∠BCA = ∠DAC (Alternate interior angle) 

∠BAC = ∠DCA (Alternate interior angle) 

AC = AC (Common) 

ABC ≅  CDA (ASA) 

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

To Prove: AB = CD, and AD = BC

Proof: As mentioned above, 

ABC ≅ CDA

Hence,

AB = CD (CPCT = Congruent Parts of Congruent Triangles) 

AD = BC (CPCT)

Theorem-3: If each pair of opposite sides of any quadrilateral is equal, then it is a parallelogram. 

To Prove: ABCD is a parallelogram.

Proof for a parallelogram

Given: AB = CD, and BC = DA

Proof: In ABD and CDB

AB = CD (Given)

BC = DA (Given)

BD = BD (Common) 

 ABD ≅  CDB (SSS)

Hence, ∠ ADB = ∠ DBC (CPCT)

And ∠ ABD = ∠CDB (CPCT)

AB is parallel to CD, and BC is parallel to AD (Inverse of Alternate interior angle theorem) 

Theorem-4: In a parallelogram, Opposite angles are equal. 

To Prove: ∠DAB = ∠DCB and ∠ADC = ∠ABC

Proving opposite angles are equal in a parallelogram

Proof: From the above equation, 

ABD ≅ CDB

Hence, ∠DAB = ∠DCB (CPCT) 

Similarly, ADC ≅ CBA 

∠ADC = ∠ABC

Theorem-5: A given quadrilateral is a parallelogram if each pair of opposite angles is equal. 

To Prove: ABCD is a parallelogram.

Proof: aA = aC and aB = aD 

Proof that the quadrilateral is a parallelogram

Solution: By angle sum property of a quadrilateral, 

∠A + ∠B + ∠C +∠D = 360 

∠A + ∠B + ∠A + ∠B = 360 

∠A + ∠B = 180 

In maths, the sum of the adjacent angles of a parallelogram is always equal to 180 degrees. Hence, ABCD is a parallelogram. 

Theorem- 6: The diagonals of a parallelogram bisect each other. 

To Prove: OD = OB and OC = OA

Diagonals of a parallelogram bisecting each other

Given: ABCD is a parallelogram.

Proof: In AOD and COB

AD = CB (Opp. sides of a parallelogram) 

∠DAO = ∠BCO (Alternate Interior angle) 

∠ADO = ∠CBO (Alternate Interior angle) 

AOD  ≅  COB (ASA)

OD = OB and OC = OA (CPCT) 

Theorem-7: If the diagonals of a parallelogram bisect each other, then it is a parallelogram. 

To Prove: ABCD is a parallelogram.

Bisecting diagonal in a parallelogram

Given: OD = OB and OC = OA

Proof: In AOD and COB

OD = OB  

OC = OA

∠AOD = ∠COB (Vertically opposite angle) 

AOD ≅  COB (SAS) 

∠DAO = ∠BCO (CPCT)

∠ADO = ∠CBO (CPCT)

Hence, ABCD is a parallelogram by the inverse of the alternate interior angle theorem. 

Theorem-8: Midpoint Theorem: The line segment joining the midpoints of any 2 sides of a triangle is parallel to the third side. 

To Prove: DE parallel to BC and DE = ½ BC 

Midpoint Theorem

Given: E and D are midpoints of AC and AB, respectively.

Construction: Draw CF parallel to AB and extend DE to point F. 

Proof: In ADE and CFE

∠ 3 = ∠ 4 (Alternate interior angle) 

∠ 1 = ∠ 2 (Vertically opposite angle) 

AE = CE (E is the midpoint of AC) 

ADE ≅   CFE (ASA)

DE = EF (CPCT) 

AD = CF (CPCT) 

AD = BD (D is the midpoint of AB) 

Hence, BD = CF 

CF is parallel to BD by construction. Hence, ABCD is a parallelogram. Meaning, 

DE is parallel to BC. 

DF = BC (Opposite sides of parallelogram) 

DE = ½ DF (as DE = EF) 

DE = ½ BC. 

Theorem-9: The line drawn through the mid-point of one side of a triangle, parallel to another side, bisects the third side. (Inverse of midpoint theorem) 

To Prove: F is the midpoint of AC. or AF=CF.

Midpoint theorem

Given: E is the midpoint of AB, and EF is parallel to BC. 

Construction: Extend EF to point D. Construct CM parallel to AB.

Solution:

ED parallel to BC and CD parallel to BE. Hence, BCDE is a parallelogram. 

AE = BE (E is the midpoint of AB) 

BE = CD (Oppo. sides of a parallelogram) 

Hence, AE = CD 

∠AFE = ∠CFD (Vertically opposite angle) 

∠EAF = ∠DCF (Alternate interior angle) 

AEF ≅   CDF (AAS)

AF = CF (CPCT) 

Also Read: Understanding Quadrilaterals

4.0Key Features of CBSE Maths Notes for Class 9 Chapter 8

  • These notes are aligned with the latest CBSE curriculum. 
  • Visual aid is provided to get a better understanding of the chapter. 
  • A step-by-step guide for solving complex problems.

Chapter-wise CBSE Notes for Class 9 Maths:

Class 9 Maths Chapter 1 - Number Systems Notes

Class 9 Maths Chapter 2 - Polynomial Notes

Class 9 Maths Chapter 3 - Coordinate Geometry Notes

Class 9 Maths Chapter 4 - Linear Equation In Two Variables Notes

Class 9 Maths Chapter 5 - Introduction To Euclids Geometry Notes

Class 9 Maths Chapter 6 - Lines and Angles Notes

Class 9 Maths Chapter 7 - Triangles Notes

Class 9 Maths Chapter 8 - Quadrilaterals Notes

Class 9 Maths Chapter 9 - Circles Notes

Class 9 Maths Chapter 10 - Herons Formula Notes

Class 9 Maths Chapter 11 - Surface Areas and Volumes Notes

Class 9 Maths Chapter 12 - Statistics Notes


Chapter-wise NCERT Solutions for Class 9 Maths:

Chapter 1: Number Systems

Chapter 2: Polynomials

Chapter 3: Coordinate Geometry

Chapter 4: Linear Equations in Two Variables

Chapter 5: Introduction to Euclid’s Geometry

Chapter 6: Lines and Angles

Chapter 7: Triangles

Chapter 8: Quadrilaterals

Chapter 9: Circles

Chapter 10: Heron’s Formula

Chapter 11: Surface Areas and Volumes

Chapter 12: Statistics


Frequently Asked Questions

In a parallelogram, the diagonals bisect each other.

A quadrilateral is a parallelogram if opposite sides are equal and parallel.

In Maths, A rectangle is a parallelogram with right angles.

The theorem about diagonals bisecting each other helps one to prove if a given quadrilateral is a parallelogram.

Related Article:-

CBSE Class 9

Students should be aware of preparation tips and syllabi when preparing for CBSE Class 9. Keep reading for more information on the CBSE 9th exam details.

CBSE Class 9 Maths

In Class 9, students learn foundational concepts that are vital for advanced studies and real-world problem-solving.

CBSE Class 9 Exam Pattern

The CBSE Class 9 exams are extremely crucial as they form the gateway to K12 Science and Math. Hence, understanding this pattern becomes important to ensure effective preparation.

CBSE Class 9 Science

A student's academic journey takes a significant turn in Class 9 Science, where they build the foundation for more advanced studies and expand their knowledge of important scientific concepts.

CBSE Sample Papers

If you are preparing for the Class 11 and 12 exams, it is essential to use CBSE Sample Papers. These papers are crucial tools that help students prepare effectively for their board exams. The CBSE releases sample papers every year in September.

CBSE Question Papers

After mastering the concepts through NCERT, it's essential to practice using CBSE Question Papers to evaluate your understanding and enhance exam readiness. These papers allow for effective self-assessment, helping you gauge how many questions may come from each chapter.

CBSE Science Topics

Science is, therefore, a systematic enterprise that organizes and builds testable explanations and...

CBSE Maths Topics

Mathematics is the study of patterns, structure, and relationships, rooted in fundamental practices like counting,...

Join ALLEN!

(Session 2025 - 26)


Choose class
Choose your goal
Preferred Mode
Choose State
  • About
    • About us
    • Blog
    • News
    • MyExam EduBlogs
    • Privacy policy
    • Public notice
    • Careers
    • Dhoni Inspires NEET Aspirants
    • Dhoni Inspires JEE Aspirants
  • Help & Support
    • Refund policy
    • Transfer policy
    • Terms & Conditions
    • Contact us
  • Popular goals
    • NEET Coaching
    • JEE Coaching
    • 6th to 10th
  • Courses
    • Online Courses
    • Distance Learning
    • Online Test Series
    • International Olympiads Online Course
    • NEET Test Series
    • JEE Test Series
    • JEE Main Test Series
    • CUET Test Series
  • Centers
    • Kota
    • Bangalore
    • Indore
    • Delhi
    • More centres
  • Exam information
    • JEE Main
    • JEE Advanced
    • NEET UG
    • CBSE
    • NCERT Solutions
    • NEET Mock Test
    • CUET
    • Olympiad
    • NEET 2025 Answer Key

ALLEN Career Institute Pvt. Ltd. © All Rights Reserved.

ISO