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CBSE Case Study Questions Class 9 Maths Chapter 8 Quadrilaterals PDF Download

CBSE Case Study Questions Class 9 Maths Chapter 8  are very important to solve for your exam. Class 9 Maths Chapter 8 Case Study Questions have been prepared for the latest exam pattern. You can check your knowledge by solving case study-based questions for Class 9 Case Study Questions Maths Chapter 8 Quadrilaterals

CBSE Case Study Questions Class 9 Maths Chapter 8

Case Study/Passage-Based Questions

Case Study 1. Laveena’s class teacher gave students some colorful papers in the shape of quadrilaterals. She asked students to make a parallelogram from it using paper folding. Laveena made the following parallelogram.

How can a parallelogram be formed by using paper folding? (a) Joining the sides of quadrilateral (b) Joining the mid-points of sides of quadrilateral (c) Joining the various quadrilaterals (d) None of these

Answer: (b) Joining the mid-points of sides of quadrilateral

Which of the following is true? (a) PQ = BD (b) PQ = 1/2 BD (c) 3PQ = BD (d) PQ = 2BD

Answer: (b) PQ = 1/2 BD

Which of the following is correct combination? (a) 2RS = BD (b) RS = 1/3 BD (c) RS = BD (d) RS = 2BD

Answer: (a) 2RS = BD

Which of the following is correct? (a) SR = 2PQ (b) PQ = SR (c) SR = 3PQ (d) SR = 4PQ

Answer: (b) PQ = SR

Case Study/Passage Based Questions

Case Study 2. Anjali and Meena were trying to prove mid-point theorem. They draw a triangle ABC, where D and E are found to be the midpoints of AB and AC respectively. DE was joined and extended to F such that DE = EF and FC is also joined.

▲ADE and ▲CFE are congruent by which criterion? (a) SSS (b) SAS (c) RHS (d) ASA

Answer: (b) SAS

∠EFC is equal to which angle? (a) ∠DAE (b) ∠EDA (c) ∠AED (d) ∠DBC

Answer: (b)∠EDA

∠ECF is equal to which angle? (a) ∠EAD (b) ∠ADE (c) ∠AED (d) ∠B

Answer: (a) ∠EAD

CF is equal to (a) EC (b) BE (c) BC (d) AD

Answer: (d) AD

CF is parallel to (a) AE (b) CE (c) BD (d) AC

Answer: (c) BD

Case Study 3. A group of students is exploring different types of quadrilaterals. They encountered the following scenario:

Four friends, Aryan, Bhavana, Chetan, and Divya, participated in a geometry project. They constructed a figure with four sides and made the following observations:

• The opposite sides of the figure are parallel.
• The opposite angles of the figure are congruent.
• The figure has two pairs of congruent adjacent sides.
• The sum of the measures of the interior angles of the figure is 360 degrees.

Based on this information, the students were asked to analyze the properties of the quadrilateral they constructed. Let’s see if you can answer the questions correctly:

MCQ Questions:

Q1. The type of quadrilateral formed by their figure is: (a) Parallelogram (b) Rhombus (c) Rectangle (d) Square

Answer: (a) Parallelogram

Q2. The measure of each angle in the figure is: (a) 90 degrees (b) 120 degrees (c) 135 degrees (d) 180 degrees

Answer: (d) 180 degrees

Q3. The figure is an example of a quadrilateral that satisfies the: (a) Opposite sides are equal condition (b) Opposite angles are congruent condition (c) Diagonals bisect each other condition (d) None of the above

Answer: (b) Opposite angles are congruent condition

Q4. The sum of the measures of the exterior angles of the figure is: (a) 90 degrees (b) 180 degrees (c) 270 degrees (d) 360 degrees

Answer: (d) 360 degrees

Q5. The figure has rotational symmetry of: (a) Order 1 (b) Order 2 (c) Order 3 (d) Order 4

Answer: (a) Order 1

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Case Study Questions Class 9 Maths Chapter 8

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CBSE Case Study Questions for Class 9 Maths Quadrilaterals Free PDF

Mere Bacchon, you must practice the CBSE Case Study Questions Class 9 Maths Quadrilaterals  in order to fully complete your preparation . They are very very important from exam point of view. These tricky Case Study Based Questions can act as a villain in your heroic exams!

I have made sure the questions (along with the solutions) prepare you fully for the upcoming exams. To download the latest CBSE Case Study Questions , just click ‘ Download PDF ’.

CBSE Case Study Questions for Class 9 Maths Quadrilaterals PDF

Checkout our case study questions for other chapters.

• Chapter 6 Lines and Angles Case Study Questions
• Chapter 7 Triangles Case Study Questions
• Chapter 9 Areas of Parallelograms and Triangles Case Study Questions
• Chapter 10 Circles Case Study Questions

How should I study for my upcoming exams?

First, learn to sit for at least 2 hours at a stretch

Solve every question of NCERT by hand, without looking at the solution.

Solve NCERT Exemplar (if available)

Sit through chapter wise FULLY INVIGILATED TESTS

Practice MCQ Questions (Very Important)

Practice Assertion Reason & Case Study Based Questions

Sit through FULLY INVIGILATED TESTS involving MCQs. Assertion reason & Case Study Based Questions

After Completing everything mentioned above, Sit for atleast 6 full syllabus TESTS.

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CBSE Class 9 Mathematics Case Study Questions

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Significance of Mathematics in Class 9

Mathematics is an important subject for students of all ages. It helps students to develop problem-solving and critical-thinking skills, and to think logically and creatively. In addition, mathematics is essential for understanding and using many other subjects, such as science, engineering, and finance.

CBSE Class 9 is an important year for students, as it is the foundation year for the Class 10 board exams. In Class 9, students learn many important concepts in mathematics that will help them to succeed in their board exams and in their future studies. Therefore, it is essential for students to understand and master the concepts taught in Class 9 Mathematics .

Case studies in Class 9 Mathematics

A case study in mathematics is a detailed analysis of a particular mathematical problem or situation. Case studies are often used to examine the relationship between theory and practice, and to explore the connections between different areas of mathematics. Often, a case study will focus on a single problem or situation and will use a variety of methods to examine it. These methods may include algebraic, geometric, and/or statistical analysis.

Example of Case study questions in Class 9 Mathematics

The Central Board of Secondary Education (CBSE) has included case study questions in the Class 9 Mathematics paper. This means that Class 9 Mathematics students will have to solve questions based on real-life scenarios. This is a departure from the usual theoretical questions that are asked in Class 9 Mathematics exams.

The following are some examples of case study questions from Class 9 Mathematics:

Class 9 Mathematics Case study question 1

There is a square park ABCD in the middle of Saket colony in Delhi. Four children Deepak, Ashok, Arjun and Deepa went to play with their balls. The colour of the ball of Ashok, Deepak,  Arjun and Deepa are red, blue, yellow and green respectively. All four children roll their ball from centre point O in the direction of   XOY, X’OY, X’OY’ and XOY’ . Their balls stopped as shown in the above image.

Answer the following questions:

Answer Key:

Class 9 Mathematics Case study question 2

• Now he told Raju to draw another line CD as in the figure
• The teacher told Ajay to mark  ∠ AOD  as 2z
• Suraj was told to mark  ∠ AOC as 4y
• Clive Made and angle  ∠ COE = 60°
• Peter marked  ∠ BOE and  ∠ BOD as y and x respectively

Now answer the following questions:

• 2y + z = 90°
• 2y + z = 180°
• 4y + 2z = 120°
• (a) 2y + z = 90°

Class 9 Mathematics Case study question 3

• (a) 31.6 m²
• (c) 513.3 m³
• (b) 422.4 m²

Class 9 Mathematics Case study question 4

How to Answer Class 9 Mathematics Case study questions

To crack case study questions, Class 9 Mathematics students need to apply their mathematical knowledge to real-life situations. They should first read the question carefully and identify the key information. They should then identify the relevant mathematical concepts that can be applied to solve the question. Once they have done this, they can start solving the Class 9 Mathematics case study question.

Students need to be careful while solving the Class 9 Mathematics case study questions. They should not make any assumptions and should always check their answers. If they are stuck on a question, they should take a break and come back to it later. With some practice, the Class 9 Mathematics students will be able to crack case study questions with ease.

Class 9 Mathematics Curriculum at Glance

At the secondary level, the curriculum focuses on improving students’ ability to use Mathematics to solve real-world problems and to study the subject as a separate discipline. Students are expected to learn how to solve issues using algebraic approaches and how to apply their understanding of simple trigonometry to height and distance problems. Experimenting with numbers and geometric forms, making hypotheses, and validating them with more observations are all part of Math learning at this level.

The suggested curriculum covers number systems, algebra, geometry, trigonometry, mensuration, statistics, graphing, and coordinate geometry, among other topics. Math should be taught through activities that include the use of concrete materials, models, patterns, charts, photographs, posters, and other visual aids.

CBSE Class 9 Mathematics (Code No. 041)

I	NUMBER SYSTEMS	10
II	ALGEBRA	20
III	COORDINATE GEOMETRY	04
IV	GEOMETRY	27
V	MENSURATION	13
VI	STATISTICS & PROBABILITY	06

Class 9 Mathematics question paper design

The CBSE Class 9 mathematics question paper design is intended to measure students’ grasp of the subject’s fundamental ideas. The paper will put their problem-solving and analytical skills to the test. Class 9 mathematics students are advised to go through the question paper pattern thoroughly before they start preparing for their examinations. This will help them understand the paper better and enable them to score maximum marks. Refer to the given Class 9 Mathematics question paper design.

QUESTION PAPER DESIGN (CLASS 9 MATHEMATICS)

1.	Exhibit memory of previously learned material by recalling facts, terms, basic concepts, and answers.  Demonstrate understanding of facts and ideas by organizing, comparing, translating, interpreting, giving descriptions, and stating main ideas	43	54
2.	Solve problems to new situations by applying acquired knowledge, facts, techniques and rules in a different way.	19	24
3.	Examine and break information into parts by identifying motives or causes. Make inferences and find evidence to support generalizations Present and defend opinions by making judgments about information, validity of ideas, or quality of work based on a set of criteria. Compile information together in a different way by combining elements in a new pattern or proposing alternative solutions	18	22
		80	100

myCBSEguide: Blessing in disguise

Class 9 is an important milestone in a student’s life. It is the last year of high school and the last chance to score well in the CBSE board exams. myCBSEguide is the perfect platform for students to get started on their preparations for Class 9 Mathematics. myCBSEguide provides comprehensive study material for all subjects, including practice questions, sample papers, case study questions and mock tests. It also offers tips and tricks on how to score well in exams. myCBSEguide is the perfect door to enter for class 9 CBSE preparations.

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16 thoughts on “CBSE Class 9 Mathematics Case Study Questions”

This method is not easy for me

aarti and rashika are two classmates. due to exams approaching in some days both decided to study together. during revision hour both find difficulties and they solved each other’s problems. aarti explains simplification of 2+ ?2 by rationalising the denominator and rashika explains 4+ ?2 simplification of (v10-?5)(v10+ ?5) by using the identity (a – b)(a+b). based on above information, answer the following questions: 1) what is the rationalising factor of the denominator of 2+ ?2 a) 2-?2 b) 2?2 c) 2+ ?2 by rationalising the denominator of aarti got the answer d) a) 4+3?2 b) 3+?2 c) 3-?2 4+ ?2 2+ ?2 d) 2-?3 the identity applied to solve (?10-?5) (v10+ ?5) is a) (a+b)(a – b) = (a – b)² c) (a – b)(a+b) = a² – b² d) (a-b)(a+b)=2(a² + b²) ii) b) (a+b)(a – b) = (a + b

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• NCERT Solutions
• NCERT Class 9
• NCERT 9 Maths
• Chapter 8: Quadrilaterals

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals are an educational aid for students to solve and learn simple and difficult problems. It includes a complete set of questions organised with an advanced level of difficulty, which provides students ample opportunity to apply their knowledge and skills. Get free NCERT Solution s for Class 9 Maths Chapter 8 Quadrilaterals devised according to the latest update on CBSE Syllabus for 2023-24.

Download Exclusively Curated Chapter Notes for Class 9 Maths Chapter – 8 Quadrilaterals

Download most important questions for class 9 maths chapter – 8 quadrilaterals.

The NCERT Solutions will help the students to understand the concept of Quadrilaterals – mainly the basics, properties, and some important theorems. The Class 9 Maths NCERT solutions will not only help students to clear their doubts but also prepare more efficiently for the CBSE examination.

• Chapter 1 Number System
• 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 Areas of Parallelograms and Triangles
• Chapter 10 Circles
• Chapter 11 Constructions
• Chapter 12 Heron’s Formula
• Chapter 13 Surface Areas and Volumes
• Chapter 14 Statistics
• Chapter 15 Introduction to Probability

NCERT Solutions for Class 9 Maths Chapter 8 – Quadrilaterals

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Access Answers to NCERT Class 9 Maths Chapter 8 – Quadrilaterals

Exercise 8.1 page: 146.

1. The angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.

Let the common ratio between the angles be x.

We know that the sum of the interior angles of the quadrilateral = 360°

3x+5x+9x+13x = 360°

⇒ 30x = 360°

, Angles of the quadrilateral are:

3x = 3×12° = 36°

5x = 5×12° = 60°

9x = 9×12° = 108°

13x = 13×12° = 156°

2. If the diagonals of a parallelogram are equal, then show that it is a rectangle.

Given that,

To show that ABCD is a rectangle if the diagonals of a parallelogram are equal

To show ABCD is a rectangle, we have to prove that one of its interior angles is right-angled.

In ΔABC and ΔBAD,

AB = BA (Common)

BC = AD (Opposite sides of a parallelogram are equal)

AC = BD (Given)

∠A+∠B = 180° (Sum of the angles on the same side of the transversal)

⇒ 2∠A = 180°

⇒ ∠A = 90° = ∠B

Therefore, ABCD is a rectangle.

Hence Proved.

3. Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

Let ABCD be a quadrilateral whose diagonals bisect each other at right angles.

and ∠AOB = ∠BOC = ∠OCD = ∠ODA = 90°

To show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus, we have to prove that ABCD is a parallelogram and AB = BC = CD = AD

In ΔAOB and ΔCOB,

OA = OC (Given)

∠AOB = ∠COB (Opposite sides of a parallelogram are equal)

OB = OB (Common)

Similarly, we can prove,

, AB = BC = CD = AD

Opposite sides of a quadrilateral are equal. Hence, it is a parallelogram.

ABCD is rhombus as it is a parallelogram whose diagonals intersect at a right angle.

4. Show that the diagonals of a square are equal and bisect each other at right angles.

Let ABCD be a square and its diagonals AC and BD intersect each other at O.

To show that,

and ∠AOB = 90°

∠ABC = ∠BAD = 90°

BC = AD (Given)

diagonals are equal.

In ΔAOB and ΔCOD,

∠BAO = ∠DCO (Alternate interior angles)

∠AOB = ∠COD (Vertically opposite)

AB = CD (Given)

AO = CO [CPCT].

, Diagonal bisect each other.

OB = OB (Given)

AO = CO (diagonals are bisected)

AB = CB (Sides of the square)

also, ∠AOB = ∠COB

∠AOB+∠COB = 180° (Linear pair)

Thus, ∠AOB = ∠COB = 90°

, Diagonals bisect each other at right angles

5. Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

Let ABCD be a quadrilateral and its diagonals AC and BD bisect each other at a right angle at O.

To prove that,

The Quadrilateral ABCD is a square.

AO = CO (Diagonals bisect each other)

OB = OD (Diagonals bisect each other)

AB = CD [CPCT] — (i)

∠OAB = ∠OCD (Alternate interior angles)

In ΔAOD and ΔCOD,

∠AOD = ∠COD (Vertically opposite)

OD = OD (Common)

AD = CD [CPCT] — (ii)

AD = BC and AD = CD

⇒ AD = BC = CD = AB — (ii)

and ∠ADC+∠BCD = 180° (co-interior angles)

⇒ 2∠ADC = 180°

⇒∠ADC = 90° — (iii)

One of the interior angles is a right angle.

Thus, from (i), (ii) and (iii), given quadrilateral ABCD is a square.

6. Diagonal AC of a parallelogram ABCD bisects ∠A (see Fig. 8.19). Show that

(i) it bisects ∠C also,

(ii) ABCD is a rhombus.

(i) In ΔADC and ΔCBA,

AD = CB (Opposite sides of a parallelogram)

DC = BA (Opposite sides of a parallelogram)

AC = CA (Common Side)

∠ACD = ∠CAB by CPCT

and ∠CAB = ∠CAD (Given)

⇒ ∠ACD = ∠BCA

AC bisects ∠C also.

(ii) ∠ACD = ∠CAD (Proved above)

⇒ AD = CD (Opposite sides of equal angles of a triangle are equal)

Also, AB = BC = CD = DA (Opposite sides of a parallelogram)

ABCD is a rhombus.

7. ABCD is a rhombus. Show that diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B as well as ∠D.

AC and BD are its diagonals.

AD = CD (Sides of a rhombus)

∠DAC = ∠DCA (Angles opposite of equal sides of a triangle are equal.)

also, AB || CD

⇒∠DAC = ∠BCA (Alternate interior angles)

⇒∠DCA = ∠BCA

, AC bisects ∠C.

We can prove that diagonal AC bisects ∠A.

Following the same method,

We can prove that the diagonal BD bisects ∠B and ∠D.

8. ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C. Show that:

(i) ABCD is a square

(ii) Diagonal BD bisects ∠B as well as ∠D.

(i) ∠DAC = ∠DCA (AC bisects ∠A as well as ∠C)

⇒ AD = CD (Sides opposite to equal angles of a triangle are equal)

also, CD = AB (Opposite sides of a rectangle)

,AB = BC = CD = AD

Thus, ABCD is a square.

(ii) In ΔBCD,

⇒ ∠CDB = ∠CBD (Angles opposite to equal sides are equal)

also, ∠CDB = ∠ABD (Alternate interior angles)

⇒ ∠CBD = ∠ABD

Thus, BD bisects ∠B

∠CBD = ∠ADB

⇒ ∠CDB = ∠ADB

Thus, BD bisects ∠B as well as ∠D.

9. In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that:

(i) ΔAPD ≅ ΔCQB

(ii) AP = CQ

(iii) ΔAQB ≅ ΔCPD

(iv) AQ = CP

(v) APCQ is a parallelogram

(i) In ΔAPD and ΔCQB,

DP = BQ (Given)

∠ADP = ∠CBQ (Alternate interior angles)

AD = BC (Opposite sides of a parallelogram)

(ii) AP = CQ by CPCT as ΔAPD ≅ ΔCQB.

(iii) In ΔAQB and ΔCPD,

BQ = DP (Given)

∠ABQ = ∠CDP (Alternate interior angles)

AB = CD (Opposite sides of a parallelogram)

(iv) As ΔAQB ≅ ΔCPD

(v) From the questions (ii) and (iv), it is clear that APCQ has equal opposite sides and also has equal and opposite angles. , APCQ is a parallelogram.

10. ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see Fig. 8.21). Show that

(i) ΔAPB ≅ ΔCQD

(i) In ΔAPB and ΔCQD,

∠ABP = ∠CDQ (Alternate interior angles)

∠APB = ∠CQD (= 90 o as AP and CQ are perpendiculars)

AB = CD (ABCD is a parallelogram)

(ii) As ΔAPB ≅ ΔCQD.

11. In ΔABC and ΔDEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, B and C are joined to vertices D, E and F, respectively (see Fig. 8.22).

(i) quadrilateral ABED is a parallelogram

(ii) quadrilateral BEFC is a parallelogram

(iii) AD || CF and AD = CF

(iv) quadrilateral ACFD is a parallelogram

(v) AC = DF

(vi) ΔABC ≅ ΔDEF.

(i) AB = DE and AB || DE (Given)

Two opposite sides of a quadrilateral are equal and parallel to each other.

Thus, quadrilateral ABED is a parallelogram

(ii) Again BC = EF and BC || EF.

Thus, quadrilateral BEFC is a parallelogram.

(iii) Since ABED and BEFC are parallelograms.

⇒ AD = BE and BE = CF (Opposite sides of a parallelogram are equal)

Also, AD || BE and BE || CF (Opposite sides of a parallelogram are parallel)

(iv) AD and CF are opposite sides of quadrilateral ACFD which are equal and parallel to each other. Thus, it is a parallelogram.

(v) Since ACFD is a parallelogram

AC || DF and AC = DF

(vi) In ΔABC and ΔDEF,

AB = DE (Given)

BC = EF (Given)

AC = DF (Opposite sides of a parallelogram)

12. ABCD is a trapezium in which AB || CD and AD = BC (see Fig. 8.23). Show that

(i) ∠A = ∠B

(ii) ∠C = ∠D

(iii) ΔABC ≅ ΔBAD

(iv) diagonal AC = diagonal BD

[Hint: Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]

To Construct: Draw a line through C parallel to DA intersecting AB produced at E.

(i) CE = AD (Opposite sides of a parallelogram)

AD = BC (Given)

⇒∠CBE = ∠CEB

∠A+∠CBE = 180° (Angles on the same side of transversal and ∠CBE = ∠CEB)

∠B +∠CBE = 180° ( As Linear pair)

(ii) ∠A+∠D = ∠B+∠C = 180° (Angles on the same side of transversal)

⇒∠A+∠D = ∠A+∠C (∠A = ∠B)

(iii) In ΔABC and ΔBAD,

AB = AB (Common)

∠DBA = ∠CBA

(iv) Diagonal AC = diagonal BD by CPCT as ΔABC ≅ ΔBAD.

Exercise 8.2 Page: 150

1. ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig 8.29). AC is a diagonal. Show that: (i) SR || AC and SR = 1/2 AC (ii) PQ = SR (iii) PQRS is a parallelogram.

(i) In ΔDAC,

R is the mid point of DC and S is the mid point of DA.

Thus by mid point theorem, SR || AC and SR = ½ AC

(ii) In ΔBAC,

P is the mid point of AB and Q is the mid point of BC.

Thus by mid point theorem, PQ || AC and PQ = ½ AC

also, SR = ½ AC

(iii) SR || AC ———————- from question (i)

and, PQ || AC ———————- from question (ii)

⇒ SR || PQ – from (i) and (ii)

also, PQ = SR

, PQRS is a parallelogram.

2. ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA, respectively. Show that the quadrilateral PQRS is a rectangle.

Given in the question,

ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA, respectively.

PQRS is a rectangle.

Construction,

Join AC and BD.

In ΔDRS and ΔBPQ,

DS = BQ (Halves of the opposite sides of the rhombus)

∠SDR = ∠QBP (Opposite angles of the rhombus)

DR = BP (Halves of the opposite sides of the rhombus)

RS = PQ [CPCT]———————- (i)

In ΔQCR and ΔSAP,

RC = PA (Halves of the opposite sides of the rhombus)

∠RCQ = ∠PAS (Opposite angles of the rhombus)

CQ = AS (Halves of the opposite sides of the rhombus)

RQ = SP [CPCT]———————- (ii)

R and Q are the mid points of CD and BC, respectively.

P and S are the mid points of AD and AB, respectively.

also, ∠PQR = 90°

RS = PQ and RQ = SP from (i) and (ii)

, PQRS is a rectangle.

3. ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA, respectively. Show that the quadrilateral PQRS is a rhombus.

ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA, respectively.

PQRS is a rhombus.

P and Q are the mid-points of AB and BC, respectively

, PQ || AC and PQ = ½ AC (Midpoint theorem) — (i)

SR || AC and SR = ½ AC (Midpoint theorem) — (ii)

So, PQ || SR and PQ = SR

As in quadrilateral PQRS one pair of opposite sides is equal and parallel to each other, so, it is a parallelogram.

, PS || QR and PS = QR (Opposite sides of parallelogram) — (iii)

Q and R are mid points of side BC and CD, respectively.

, QR || BD and QR = ½ BD (Midpoint theorem) — (iv)

AC = BD (Diagonals of a rectangle are equal) — (v)

From equations (i), (ii), (iii), (iv) and (v),

PQ = QR = SR = PS

So, PQRS is a rhombus.

Hence Proved

4. ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F (see Fig. 8.30). Show that F is the mid-point of BC.

ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the mid-point of AD.

F is the mid-point of BC.

BD intersected EF at G.

E is the mid point of AD and also EG || AB.

Thus, G is the mid point of BD (Converse of mid point theorem)

G is the mid point of BD and also GF || AB || DC.

Thus, F is the mid point of BC (Converse of mid point theorem)

5. In a parallelogram ABCD, E and F are the mid-points of sides AB and CD, respectively (see Fig. 8.31). Show that the line segments AF and EC trisect the diagonal BD.

ABCD is a parallelogram. E and F are the mid-points of sides AB and CD, respectively.

AF and EC trisect the diagonal BD.

ABCD is a parallelogram

also, AE || FC

AB = CD (Opposite sides of parallelogram ABCD)

⇒½ AB = ½ CD

⇒ AE = FC (E and F are midpoints of side AB and CD)

AECF is a parallelogram (AE and CF are parallel and equal to each other)

AF || EC (Opposite sides of a parallelogram)

F is mid point of side DC and FP || CQ (as AF || EC).

P is the mid-point of DQ (Converse of mid-point theorem)

⇒ DP = PQ — (i)

E is midpoint of side AB and EQ || AP (as AF || EC).

Q is the mid-point of PB (Converse of mid-point theorem)

⇒ PQ = QB — (ii)

From equations (i) and (i),

DP = PQ = BQ

Hence, the line segments AF and EC trisect the diagonal BD.

6 . Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other. Solution:

Let ABCD be a quadrilateral and P, Q, R and S the mid points of AB, BC, CD and DA, respectively.

R and S are the mid points of CD and DA, respectively.

, SR || AC.

Similarly we can show that,

PS || BD and

, PQRS is parallelogram.

PR and QS are the diagonals of the parallelogram PQRS. So, they will bisect each other.

7. ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that (i) D is the mid-point of AC (ii) MD ⊥ AC (iii) CM = MA = ½ AB

(i) In ΔACB,

M is the midpoint of AB and MD || BC

, D is the midpoint of AC (Converse of mid point theorem)

(ii) ∠ACB = ∠ADM (Corresponding angles)

also, ∠ACB = 90°

, ∠ADM = 90° and MD ⊥ AC

(iii) In ΔAMD and ΔCMD,

AD = CD (D is the midpoint of side AC)

∠ADM = ∠CDM (Each 90°)

DM = DM (common)

also, AM = ½ AB (M is midpoint of AB)

Hence, CM = MA = ½ AB

NCERT Solutions for Class 9 Maths Chapter 8 explains the Angle Sum Property of a Quadrilateral, Types of Quadrilaterals and Mid-Point theorem.

Topics covered in this chapter help the students understand the basics of a quadrilateral geometrical figure, its properties and various important theorems. This chapter of NCERT Solutions for Class 9 Maths is extremely crucial as the formulas and theorem results are extensively used in several other maths concepts in higher grades.

Chapter 8 Quadrilaterals is included in the CBSE Syllabus 2023-24 and is a part of Unit – Geometry, which holds 28 marks of weightage in the CBSE Class 9 Maths exams. Two or three questions are asked every year in the board examination from this chapter.

NCERT Solutions For Class 9 Maths Chapter 8 Exercises: Get detailed solutions for all the questions listed under the below exercises: Exercise 8.1 Solutions (12 Questions) Exercise 8.2 Solutions (7 Questions) NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals NCERT Solutions for Class 9 Maths Chapter 8 is about Theorems and properties on Quadrilaterals. They are accompanied with explanatory figures and solved examples, which are explained comprehensively. The main topics covered in this chapter include:

8.1	Introduction
8.2	Angle Sum Property of a Quadrilateral
8.3	Types of Quadrilaterals
8.4	Properties of a Parallelogram
8.5	Another Condition for a Quadrilateral to be a Parallelogram
8.6	The Mid-point Theorem
8.7	Summary

Key Features of NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

• NCERT solutions have been prepared in a logical and simple language.
• Pictorial presentation of all the questions.
• Emphasizes that learning should be activity-based and knowledge-driven.
• The solutions are explained in a well-organised way.
• Step-by-step approach used to solve all NCERT questions.

Disclaimer:

Dropped Topics – 8.1 Introduction, 8.2 Angle sum property of a quadrilateral, 8.3 Types of quadrilaterals and 8.5 Another condition for a Quadrilateral to be a parallelogram.

Frequently Asked Questions on NCERT Solutions for Class 9 Maths Chapter 8

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NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

• NCERT Solutions
• Chapter 8 Quadrilaterals

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals - FREE PDF Download

In chapter 8 class 9 maths, Quadrilaterals is a fundamental topic that explores the properties and types of quadrilaterals. This chapter is crucial for understanding various geometric shapes like parallelograms, rectangles, and squares. The focus is on learning how to identify and prove the properties of these shapes, which is essential for solving related problems.

Important concepts in class 9 quadrilaterals include the properties of parallelograms, the Midpoint Theorem, and the criteria for a quadrilateral to be a parallelogram. Vedantu's Class 9 Maths NCERT Solutions provide detailed explanations and step-by-step methods to help students grasp these concepts effectively. By focusing on these key areas, students can build a strong foundation in geometry.

Glance on Maths Chapter 8 Class 9 - Quadrilaterals

Chapter 8 of Class 9  Maths deals with the properties of parallelograms, the Midpoint Theorem, and the criteria for a quadrilateral to be a parallelogram

A parallelogram is a special type of quadrilateral(has 4 sides) where opposite sides are parallel and equal. Understanding the properties and theorems related to parallelograms is crucial for solving various class 9 maths chapter 8 solutions.

In a parallelogram, both pairs of opposite sides are equal in length and parallel to each other ie, If $ABCD$ is a parallelogram, then $AB \parallel CD$ and $AD \parallel BC$, and $AB = CD$ and $AD = BC$.

The opposite angles of a parallelogram are equal ie, If $ABCD$ is a parallelogram, then $\angle A = \angle C$ and$\angle B = \angle D$.

Any two adjacent angles in a parallelogram are supplementary, meaning their sum is 180 degrees

The Midpoint Theorem states that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long.

For example, Consider a triangle $\Delta ABC $ with D and E as a midpoints of AB and AC REspectively then:

$DE\parallel BC $

$DE = \frac{1}{2} BC $

This article contains chapter notes, important questions, exemplar solutions, exercises, and video links for Chapter 8- Quadrilaterals, which you can download as PDFs.

There are two exercises (13 fully solved questions) in class 9 maths chapter 8 Quadrilaterals.

Access Exercise wise NCERT Solutions for Chapter 8 Maths Class 9

Current Syllabus Exercises of Class 9 Maths Chapter 8

Exercises Under NCERT Solutions for Class 9 Maths Chapter 8 - Quadrilaterals

NCERT Solutions for Maths Class 9 Chapter 8, "Quadrilaterals" comprises two exercises with a total of 19 questions. Here's a detailed explanation of the types of questions included in each exercise:

Exercise 8.1:

This exercise consists of 12 questions that cover a wide range of concepts related to quadrilaterals. The questions require students to identify and recognize the properties of different types of quadrilaterals. Here are the different types of questions you can expect to find in this exercise:

Identification of Quadrilaterals: In this type of question, students are given a diagram of a quadrilateral and are asked to identify the type of quadrilateral, such as a square, rectangle, parallelogram, or trapezium.

Checking Properties: In these questions, students need to confirm a given characteristic of a quadrilateral. For instance, they might be asked to confirm that the opposite sides of a parallelogram are equal.

Application of Properties: In this type of question, students need to apply the properties of quadrilaterals to solve problems. For instance, they may be asked to find the perimeter or area of a given quadrilateral.

Exercise 8.2:

This section has seven questions that emphasize applying the properties of quadrilaterals. The questions are more intricate, demanding a thorough grasp of the concepts from the chapter. Here are the types of questions you can anticipate in this exercise:

Proving Properties: In these questions, students are asked to prove a given property of a quadrilateral using the properties they have learned in the chapter.

Applying Properties: Similar to Exercise 8.1, these questions need students to use quadrilateral properties to solve problems. However, the questions in this exercise are more intricate, demanding a deeper understanding of the concepts.

Construction of Quadrilaterals: In some questions, students are asked to construct a quadrilateral based on certain given conditions, such as the length of the sides or the angles of the quadrilateral.

Access NCERT Solutions for Class 9 Maths Chapter 8– Quadrilaterals

Exercise 8.1.

1. If the diagonals of a parallelogram are equal, then show that it is a rectangle.

Given: Diagonals of the parallelogram are the same.

To prove: It is a rectangle.

Consider ABCD be the given parallelogram.

Now we need to show that ABCD is a rectangle, by proving that one of its interior angles is .

In \[\Delta ABC\] and \[\Delta DCB\] ,

AB = DC (side opposite to the parallelogram are equal)

BC = BC (in common)

AC = DB (Given)

\[\therefore \Delta ABC \cong \Delta DCB\] (By SSS Congruence rule)

\[ \Rightarrow \angle ABC{\text{ }} = \angle DCB\]

The sum of the measurements of angles on the same side of a transversal is known to be \[{180^o}\] .

Hence, ABCD is a rectangle because it is a parallelogram with a \[{90^o}\] inner angle.

2. Show that the diagonals of a square are equal and bisect each other at right angles.

Given: A square is given.

To find: The diagonals of a square are the same and bisect each other at ${90^o}$

Consider ABCD be a square.

Consider the diagonals AC and BD intersect each other at a point O.

We must first show that the diagonals of a square are equal and bisect each other at right angles,

\[{\text{AC  =  BD, OA  =  OC, OB  =  OD}}\] , and .

\[AB{\text{ }} = {\text{ }}DC\] (Sides of the square are equal)

\[\angle ABC{\text{ }} = \angle DCB\] (All the interior angles are of the value \[{90^o}\] )

\[BC{\text{ }} = {\text{ }}CB\] (Common side)

\[\therefore \Delta ABC \cong \Delta DCB\] (By SAS congruency)

\[\therefore AC{\text{ }} = {\text{ }}DB\] (By CPCT)

Hence, the diagonals of a square are equal in length.

In \[\Delta AOB\] and \[\Delta COD\] ,

\[\angle AOB{\text{ }} = \angle COD\] (Vertically opposite angles)

\[\angle ABO{\text{ }} = \angle CDO\] (Alternate interior angles)

AB = CD (Sides of a square are always equal)

\[\therefore \Delta AOB \cong \Delta COD\] (By AAS congruence rule)

\[\therefore AO{\text{ }} = {\text{ }}CO\] and \[OB{\text{ }} = {\text{ }}OD\] (By CPCT)

As a result, the diagonals of a square are bisected.

In \[\Delta AOB\] and \[\Delta COB\] ,

Because we already established that diagonals intersect each other,

\[AO{\text{ }} = {\text{ }}CO\]

\[AB{\text{ }} = {\text{ }}CB\] (Sides of a square are equal)

\[BO{\text{ }} = {\text{ }}BO\] (Common)

\[\therefore \Delta AOB \cong \Delta COB\] (By SSS congruency)

\[\therefore \angle AOB{\text{ }} = \angle COB\] (By CPCT)

However,  (Linear pair)

As a result, the diagonals of a square are at right angles to each other.

3. Diagonal AC of a parallelogram ABCD is bisecting \[\angle A\](see the given figure). Show that

(i) It is bisecting \[\angle C\]also,

(ii) ABCD is a rhombus

Answer:  

Given: Diagonal AC of a parallelogram ABCD is bisecting \[\angle A\]

To find: (i) It is bisecting \[\angle C\] also,

 (i) ABCD is a parallelogram.

\[\angle DAC{\text{ }} = \angle BCA\] (Alternate interior angles) ... (1)

And, \[\angle BAC{\text{ }} = \angle DCA\] (Alternate interior angles) ... (2)

However, it is given that AC is bisecting  \[\angle A\] .

\[\angle DAC{\text{ }} = \angle BAC\] ... (3)

From Equations (1), (2), and (3), we obtain

\[\angle DAC{\text{ }} = \angle BCA{\text{ }} = \angle BAC{\text{ }} = \angle DCA\] ... (4)

\[\angle DCA{\text{ }} = \angle BCA\]

Hence, AC is bisecting \[\angle C\] .

(ii) From Equation (4), we obtain

\[\angle DAC{\text{ }} = \angle DCA\]

\[DA{\text{ }} = {\text{ }}DC\] (Side opposite to equal angles are equal)

However, \[DA{\text{ }} = {\text{ }}BC\] and \[AB{\text{ }} = {\text{ }}CD\] (Opposite sides of a parallelogram)

\[AB{\text{ }} = {\text{ }}BC{\text{ }} = {\text{ }}CD{\text{ }} = {\text{ }}DA\]

As a result, ABCD is a rhombus.

4. ABCD is a rectangle in which diagonal AC bisects \[\angle A\] as well as \[\angle C\]. Show that:

(i) ABCD is a square

(ii) Diagonal BD bisects \[\angle B\] as well as \[\angle D\].

Given: ABCD is a rectangle where the diagonal AC bisects \[\angle A\] as well as \[\angle C\] .

To find: (i) ABCD is a square

              (ii) Diagonal BD bisects \[\angle B\] as well as \[\angle D\] .

It is given that ABCD is a rectangle. \[\angle A{\text{ }} = \angle C\]\[\]

$ \Rightarrow \dfrac{1}{2}\angle A = \dfrac{1}{2}\angle C$ (AC bisects \[\angle A\] and \[\angle C\] )

$ \Rightarrow \angle DAC = \dfrac{1}{2}\angle DCA$

CD = DA (Sides that are opposite to the equal angles are also equal)

Also, \[DA{\text{ }} = {\text{ }}BC\] and \[AB{\text{ }} = {\text{ }}CD\] (Opposite sides of the rectangle are same)

ABCD is a rectangle with equal sides on all sides.

Hence, ABCD is a square.

(ii) Let us now join BD.

In \[\Delta BCD\] ,

\[BC{\text{ }} = {\text{ }}CD\] (Sides of a square are equal to each other)

\[\angle CDB{\text{ }} = \angle CBD\] (Angles opposite to equal sides are equal)

However, \[\angle CDB{\text{ }} = \angle ABD\] (Alternate interior angles for \[AB{\text{ }}||{\text{ }}CD\] )

\[\angle CBD{\text{ }} = \angle ABD\]

BD bisects \[\angle B.\]

Also, \[\angle CBD{\text{ }} = \angle ADB\] (Alternate interior angles for \[BC{\text{ }}||{\text{ }}AD\] )

\[\angle CDB{\text{ }} = \angle ABD\]

BD bisects \[\angle D\] and \[\angle B\] .

5. In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see the given figure). Show that:

(i) \[\Delta APD \cong \Delta CQB\]

(ii) \[AP{\text{ }} = {\text{ }}CQ\]

(iii) \[\Delta AQB \cong \Delta CPD\]

(iv) \[AQ{\text{ }} = {\text{ }}CP\]

(v) APCQ is a parallelogram

Given: A parallelogram is given.

To prove: (i) \[\Delta APD \cong \Delta CQB\]

                  (ii) \[AP{\text{ }} = {\text{ }}CQ\]

      (iii) \[\Delta AQB \cong \Delta CPD\]

      (iv) \[AQ{\text{ }} = {\text{ }}CP\]

      (v) APCQ is a parallelogram

(i) In \[\Delta APD\] and \[\Delta CQB\] ,

\[\angle ADP{\text{ }} = \angle CBQ\] (Alternate interior angles for \[BC{\text{ }}||{\text{ }}AD\] )

\[AD{\text{ }} = {\text{ }}CB\] (Opposite sides of the parallelogram ABCD)

\[DP{\text{ }} = {\text{ }}BQ\] (Given)

\[\therefore \Delta APD \cong \Delta CQB\] (Using SAS congruence rule)

(ii) As we had observed that \[\Delta APD \cong \Delta CQB\] ,

\[\therefore AP{\text{ }} = {\text{ }}CQ\] (CPCT)

(iii) In \[\Delta AQB\] and \[\Delta CPD\] ,

\[\angle ABQ{\text{ }} = \angle CDP\] (Alternate interior angles for \[AB{\text{ }}||{\text{ }}CD\] )

\[AB{\text{ }} = {\text{ }}CD\] (Opposite sides of parallelogram ABCD)

\[BQ{\text{ }} = {\text{ }}DP\] (Given)

\[\therefore \Delta AQB \cong \Delta CPD\] (Using SAS congruence rule)

(iv) Since we had observed that \[\Delta AQB \cong \Delta CPD\] ,

\[\therefore AQ{\text{ }} = {\text{ }}CP\] (CPCT)

(v) From the result obtained in (ii) and (iv),

\[AQ{\text{ }} = {\text{ }}CP\] and

\[AP{\text{ }} = {\text{ }}CQ\]

APCQ is a parallelogram because the opposite sides of the quadrilateral are equal.

6. ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (See the given figure). Show that

(i) \[\Delta APB \cong \Delta CQD\]

(i) In \[\Delta APB\] and \[\Delta CQD\] ,

\[\angle APB{\text{ }} = \angle CQD\] (Each 90°)

\[AB{\text{ }} = {\text{ }}CD\] (The opposite sides of a parallelogram ABCD)

\[\angle ABP{\text{ }} = \angle CDQ\] (Alternate interior angles for \[AB{\text{ }}||{\text{ }}CD\] )

\[\therefore \Delta APB \cong \Delta CQD\] (By AAS congruency)

(ii) By using

\[\therefore \Delta APB \cong \Delta CQD\] , we obtain

\[AP{\text{ }} = {\text{ }}CQ\] (By CPCT)

7. ABCD is a trapezium in which AB || CD and AD = BC (see the given figure). Show that

(i) ∠A = ∠B

(ii) ∠C = ∠D

(iii) \[\Delta ABC \cong \Delta BAD\]

(iv) diagonal AC = diagonal BD

(Hint: Extend AB and draw a line through C parallel to DA intersecting AB produced at E.)

Given: ABCD is a trapezium.

To find: (i) ∠A = ∠B

Let us extend AB by drawing a line through C, which is parallel to AD, intersecting AE at point

E. It is clear that AECD is a parallelogram.

(i) \[AD{\text{ }} = {\text{ }}CE\] (Opposite sides of parallelogram AECD)

However, \[AD{\text{ }} = {\text{ }}BC\] (Given)

Therefore, \[BC{\text{ }} = {\text{ }}CE\]

\[\angle CEB{\text{ }} = \angle CBE\] (Angle opposite to the equal sides are also equal)

Consideing parallel lines AD and CE. AE is the transversal line for them.

 (Angles on a same side of transversal)

 (Using the relation ∠CEB = ∠CBE) ... (1)

However,  (Linear pair angles) ... (2)

From Equations (1) and (2), we obtain

\[\angle A{\text{ }} = \angle B\]

 (ii) \[AB{\text{ }}||{\text{ }}CD\]

 (Angles on a same side of the transversal)

Also, \[\angle C{\text{ }} + \angle B{\text{ }} = {\text{ }}180^\circ \] (Angles on a same side of a transversal)

\[\therefore \angle A{\text{ }} + \angle D{\text{ }} = \angle C{\text{ }} + \angle B\]

However, \[\angle A{\text{ }} = \angle B\] (Using the result obtained in (i))

\[\therefore \angle C{\text{ }} = \angle D\]

 (iii) In \[\Delta ABC\] and \[\Delta BAD\] ,

\[AB{\text{ }} = {\text{ }}BA\] (Common side)

\[BC{\text{ }} = {\text{ }}AD\] (Given)

\[\angle B{\text{ }} = \angle A\] (Proved before)

\[\therefore \Delta ABC \cong \Delta BAD\] (SAS congruence rule)

(iv) We had seen that,

\[\Delta ABC \cong \Delta BAD\]

\[\therefore AC{\text{ }} = {\text{ }}BD\] (By CPCT)

Exercise 8.2

1. ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see the given figure). AC is a diagonal.

(i) \[SR{\text{ }}||{\text{ }}AC\] and \[SR = \dfrac{1}{2}\;AC\]

(ii) PQ = SR

(iii) PQRS is a parallelogram.

Given: ABCD is a quadrilateral

To prove: (i) \[SR{\text{ }}||{\text{ }}AC\] and \[SR = \dfrac{1}{2}\;AC\]

     (ii) PQ = SR

     (iii) PQRS is a parallelogram.

(i) In \[\Delta ADC\] , S and R are the mid-points of sides AD and CD respectively.

In a triangle, the line segment connecting the midpoints of any two sides is parallel to and half of the third side.

\[\therefore SR{\text{ }}||{\text{ }}AC\] and \[SR{\text{ }} = {\text{ }}\dfrac{1}{2}{\text{ }}AC\] ... (1)

(ii) In ∆ABC, P and Q are mid-points of sides AB and BC respectively. Therefore, by using midpoint theorem,

\[PQ{\text{ }}||{\text{ }}AC\] and \[PQ{\text{ }} = {\text{ }}\dfrac{1}{2}{\text{ }}AC\] ... (2)

Using Equations (1) and (2), we obtain

\[PQ{\text{ }}||{\text{ }}SR\] and \[PQ{\text{ }} = {\text{ }}\dfrac{1}{2}SR\] ... (3)

\[\therefore PQ{\text{ }} = {\text{ }}SR\]

 (iii) From Equation (3), we obtained

\[PQ{\text{ }}||{\text{ }}SR\] and  \[PQ{\text{ }} = {\text{ }}SR\]

Clearly, one pair of quadrilateral PQRS opposing sides is parallel and equal.

PQRS is thus a parallelogram.

2. ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.

Given: ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively.

To find: Quadrilateral PQRS is a rectangle

In \[\Delta ABC\] , P and Q are the mid-points of sides AB and BC respectively.

\[PQ{\text{ }}||{\text{ }}AC{\text{ , }}PQ{\text{ }} = {\text{ }}\dfrac{1}{2}AC\] (Using mid-point theorem) ... (1)

In \[\Delta ADC\] ,

R and S are the mid-points of CD and AD respectively.

\[RS{\text{ }}||{\text{ }}AC{\text{ , }}RS{\text{ }} = {\text{ }}\dfrac{1}{2}{\text{ }}AC\] (Using mid-point theorem) ... (2)

\[PQ{\text{ }}||{\text{ }}RS\] and \[PQ{\text{ }} = {\text{ }}RS\]

It is a parallelogram because one pair of opposing sides of quadrilateral PQRS is equal and parallel to each other. At position O, the diagonals of rhombus ABCD should cross.

In quadrilateral OMQN,

\[MQ{\text{ }}\left| {\left| {{\text{ }}ON{\text{ }}({\text{ }}PQ{\text{ }}} \right|} \right|{\text{ }}AC)\]

\[QN{\text{ }}\left| {\left| {{\text{ }}OM{\text{ }}({\text{ }}QR{\text{ }}} \right|} \right|{\text{ }}BD)\]

Hence , OMQN is a parallelogram.

\[\begin{array}{*{20}{l}} {\therefore \angle MQN{\text{ }} = \angle NOM} \\ {\therefore \angle PQR{\text{ }} = \angle NOM} \end{array}\]

Since,  \[\angle NOM{\text{ }} = {\text{ }}90^\circ \] (Diagonals of the rhombus are perpendicular to each other)

\[\therefore \angle PQR{\text{ }} = {\text{ }}90^\circ \]

Clearly, PQRS is a parallelogram having one of its interior angles as .

So , PQRS is a rectangle.

3. ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.

Given: ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively.

To prove: The quadrilateral PQRS is a rhombus.

Let us join AC and BD.

In \[\Delta ABC\] ,

P and Q are the mid-points of AB and BC respectively.

\[\therefore PQ{\text{ }}||{\text{ }}AC\] and \[PQ{\text{ }} = {\text{ }}\dfrac{1}{2}{\text{ }}AC\] (Mid-point theorem) ... (1)

Similarly in \[\Delta ADC\] ,

\[SR{\text{ }}||{\text{ }}AC{\text{ , }}SR{\text{ }} = {\text{ }}\dfrac{1}{2}{\text{ }}AC\] (Mid-point theorem) ... (2)

Clearly, \[PQ{\text{ }}||{\text{ }}SR\] and \[PQ{\text{ }} = {\text{ }}SR\]

It is a parallelogram because one pair of opposing sides of quadrilateral PQRS is equal and parallel to each other.

\[\therefore PS{\text{ }}||{\text{ }}QR{\text{ }},{\text{ }}PS{\text{ }} = {\text{ }}QR\] (Opposite sides of parallelogram) ... (3)

In \[\Delta BCD\] , Q and R are the mid-points of side BC and CD respectively.

\[\therefore QR{\text{ }}||{\text{ }}BD{\text{ , }}QR{\text{ }} = {\text{ }}\dfrac{1}{2}BD\] (Mid-point theorem) ... (4)

Also, the diagonals of a rectangle are equal.

\[\therefore AC{\text{ }} = {\text{ }}BD\] …(5)

By using Equations (1), (2), (3), (4), and (5), we obtain

\[PQ{\text{ }} = {\text{ }}QR{\text{ }} = {\text{ }}SR{\text{ }} = {\text{ }}PS\]

So , PQRS is a rhombus

4. ABCD is a trapezium in which \[AB{\text{ }}||{\text{ }}DC\], BD is a diagonal and E is the mid - point of AD. A line is drawn through E parallel to AB intersecting BC at F (see the given figure). Show that F is the mid-point of BC.

Given: ABCD is a trapezium in which \[AB{\text{ }}||{\text{ }}DC\] , BD is a diagonal and E is the mid - point of AD. A line is drawn through E parallel to AB intersecting BC at F.

To prove: F is the mid-point of BC.

Let EF intersect DB at G.

We know that a line traced through the mid-point of any side of a triangle and parallel to another side bisects the third side by the reverse of the mid-point theorem.

In \[\Delta ABD\] ,

\[EF{\text{ }}||{\text{ }}AB\] and E is the mid-point of AD.

Hence , G will be the mid-point of DB.

As \[EF{\text{ }}\left| {\left| {{\text{ }}AB{\text{ , }}AB{\text{ }}} \right|} \right|{\text{ }}CD\] ,

\[\therefore EF{\text{ }}||{\text{ }}CD\] (Two lines parallel to the same line are parallel)

In \[\Delta BCD\] , \[GF{\text{ }}||{\text{ }}CD\] and G is the mid-point of line BD. So , by using converse of mid-point

theorem, F is the mid-point of BC.

5. In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (see the given figure). Show that the line segments AF and EC trisect the diagonal BD.

Given: In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively To prove: The line segments AF and EC trisect the diagonal BD.

ABCD is a parallelogram.

\[AB{\text{ }}||{\text{ }}CD\]

And hence, \[AE{\text{ }}||{\text{ }}FC\]

Again, AB = CD (Opposite sides of parallelogram ABCD)

\[\dfrac{1}{2}AB{\text{ }} = {\text{ }}\dfrac{1}{2}CD\]

\[AE{\text{ }} = {\text{ }}FC\] (E and F are mid-points of side AB and CD)

In quadrilateral AECF, one pair of the opposite sides (AE and CF) is parallel and same to each other. So , AECF is a parallelogram.

\[\therefore AF{\text{ }}||{\text{ }}EC\] (Opposite sides of a parallelogram)

In \[\Delta DQC\] , F is the mid-point of side DC and \[FP{\text{ }}||{\text{ }}CQ\] (as \[AF{\text{ }}||{\text{ }}EC\] ). So , by using the converse of mid-point theorem, it can be said that P is the mid-point of DQ.

\[\therefore DP{\text{ }} = {\text{ }}PQ\] ... (1)

Similarly, in \[\Delta APB\] , E is the mid-point of side AB and \[EQ{\text{ }}||{\text{ }}AP\] (as \[AF{\text{ }}||{\text{ }}EC\] ).

As a result, the reverse of the mid-point theorem may be used to say that Q is the mid-point of PB.

\[\therefore PQ{\text{ }} = {\text{ }}QB\] ... (2)

From Equations (1) and (2),

\[DP{\text{ }} = {\text{ }}PQ{\text{ }} = {\text{ }}BQ\]

Hence, the line segments AF and EC trisect the diagonal BD.

6. ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that

(i) D is the mid-point of AC

(ii) MD $ \bot $ AC

(iii) \[CM{\text{ }} = {\text{ }}MA{\text{ }} = \dfrac{1}{2}AB\]

Given: ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D.

To prove: (i) D is the mid-point of AC

(i) In \[\Delta ABC\] ,

It is given that M is the mid-point of AB and \[MD{\text{ }}||{\text{ }}BC\] .

Therefore, D is the mid-point of AC. (Converse of the mid-point theorem)

(ii) As \[DM{\text{ }}||{\text{ }}CB\] and AC is a transversal line for them, therefore,

 (Co-interior angles)

(iii) Join MC.

In \[\Delta AMD\] and \[\Delta CMD\] ,

\[AD{\text{ }} = {\text{ }}CD\] (D is the mid-point of side AC)

\[\angle ADM{\text{ }} = \angle CDM\] (Each )

DM = DM (Common)

\[\therefore \Delta AMD \cong \Delta CMD\] (By SAS congruence rule)

Therefore, \[AM{\text{ }} = {\text{ }}CM\] (By CPCT)

However, \[{\text{ }}AM{\text{ }} = {\text{ }}\dfrac{1}{2}{\text{ }}AB\] (M is mid-point of AB)

Therefore, it is said that

\[CM{\text{ }} = {\text{ }}AM{\text{ }} = {\text{ }}\dfrac{1}{2}{\text{ }}AB\]

Overview of Deleted Syllabus for CBSE Class 9 Maths  Quadrilaterals

Chapter	Dropped Topics
Quadrilaterals	Introduction
	Angle sum property of a quadrilateral
	Types of quadrilaterals
	Another condition for a Quadrilateral to be a parallelogram

Class 9 Maths Chapter 8: Exercises Breakdown

Exercise	Number of Questions
Exercise 8.1	7 Questions and Solutions
Exercise 8.2	6 Questions and Solutions

Class 9 Maths Chapter 8 is an important chapter that lays the foundation for future mathematics. Vedantu's Class 9 Maths Chapter 8 Solutions is a comprehensive and informative resource that will help students understand the concepts, solve problems, and improve their analytical skills.

Success in exams requires regular practice. Vedantu's Class 9 Maths Chapter 8 Solutions offers an extensive set of practice questions along with solutions, aiding students in thorough exam preparation. Questions from Chapter 8 - Quadrilaterals in Class 9 Maths might range from 2-4 questions, including both short and long answer types.

Students can also download a free PDF of Vedantu's Class 9 Maths Chapter 8 Solutions for easy access and offline use.

CBSE Class 9 Maths Chapter 8 Other Study Materials

S. No	Important Links for Class 9 Quadrilaterals
1
2
3
4
5
6

Chapter-Specific NCERT Solutions for Class 9  Maths

Given below are the chapter-wise NCERT Solutions for Class 9 Maths. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.

NCERT Solutions Class 9 Chapter-wise Maths PDF

FAQs on NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

1. What is the relation between square, Rectangle, and Rhombus?

A Square is a Rectangle and a Rhombus. But Rhombus and Rectangle are not Squares.

2. What is the relation between a trapezium and a parallelogram?

A parallelogram is a trapezium but a trapezium is not a parallelogram. 

3. How to avoid silly mistakes?

Paying attention to the theorems and their usage in the notes provided would minimize the silly mistakes. A clear understanding of the steps involved and practice is the key. Going through the  NCERT Solutions for Class 9 Chapter 8 would clear all your dangling doubts and would learn a lot of alternative steps.

4. How to increase our score in CBSE examinations?

CBSE test papers mainly test the understanding of the students. So a clear explanation for every step would help you score more. As the Quadrilateral chapter involves long steps of proof, the reason for every conclusion has to be stated neatly and clearly. Presentation in an organized manner can give a further push to your grades.

5. What is the theorem of Chapter 8 quadrilateral?

Theorem 8.2 can be stated as given below : If a quadrilateral is a parallelogram, then each pair of its opposite sides is equal. So its converse is : Theorem 8.3 : If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.

6. What is the formula of quadrilateral ABCD?

The area of the quadrilateral ABCD = Sum of areas of ΔBCD and ΔABD. Thus, the area of the quadrilateral ABCD = (1/2) × d × h1 + (1/2) × d × h2 = (1/2) × d × (h2 + h2 ).

7 What is the importance of NCERT quadrilaterals class 9?

This chapter is crucial for understanding the properties and types of quadrilaterals, including parallelograms, rectangles, and squares.

8. What key properties of parallelograms should I focus on in NCERT quadrilaterals class 9?

Focus on properties such as opposite sides being equal and parallel, opposite angles being equal, and diagonals bisecting each other.

9. How many questions from maths quadrilateral class 9 were asked in previous year exams?

Questions from Chapter 8 - Quadrilaterals in Class 9 Maths might range from 2-4 questions, including both short and long answer types.

10. What is the Midpoint Theorem and its application quadrilaterals class 9 NCERT solutions?

The Midpoint Theorem states that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. It is used to solve problems related to quadrilaterals.

11. How can Vedantu's solutions help in quadrilaterals class 9 NCERT solutions?

Vedantu provides step-by-step explanations and detailed solutions to help students understand and solve problems effectively.

12. How do I prove that a given quadrilateral is a parallelogram in NCERT quadrilaterals class 9?

Use properties such as opposite sides being equal and parallel or diagonals bisecting each other to prove a quadrilateral is a parallelogram.

13. What are the conditions for a quadrilateral to be a parallelogram in class 9 quadrilaterals?

Conditions include both pairs of opposite sides being equal and parallel, and both pairs of opposite angles being equal.

14. What should I focus on while studying chapter 8 class 9 maths?

Focus on understanding the properties, theorems, and their applications in solving geometric problems.

15. How many types of quadrilaterals are there, and what are they?

Key types include parallelograms, rectangles, squares, rhombuses, and trapezoids. Each type has its unique properties and applications.

NCERT Solutions for Class 9 Maths

Ncert solutions for class 9.

NCERT Solutions Class 9 Maths Chapter 8 Quadrilaterals

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals where quadrilateral is a four-sided polygon with four vertices and four angles. They are further classified into various types, including rectangle , square, parallelogram, rhombus, trapezium , and kite. The most common examples of quadrilaterals in the real world are the rectangular floor tiles, laptop screens, windows, signboards, etc. Although most of the examples mentioned above can be categorized as rectangles, having a deep understanding of all the types of quadrilaterals will enable students to think through their applications. NCERT Solutions Class 9 Maths Chapter 8 is helpful for the students to understand the concept of Quadrilaterals and their basic properties. This chapter efficiently covers all the important formulas, questions, and theorems based on the angle, diagonals, the sum of angles , and length of the sides of types of quadrilaterals.

Learning the properties of quadrilaterals is helpful in finding the missing angles and sides. The concepts covered in the class 9 maths NCERT solutions chapter 8 quadrilaterals are extremely important as they form the basis of understanding many important topics in higher grades. This chapter will also enable students to explore some practical applications of quadrilaterals in detail through various examples and sample questions. To explore more, you can download the pdf files in the links below and also find some of these in the exercises given below.

• NCERT Solutions Class 9 Maths Chapter 8 Ex 8.1
• NCERT Solutions Class 9 Maths Chapter 8 Ex 8.2

NCERT Solutions for Class 9 Maths Chapter 8 PDF

NCERT solutions maths for class 9 chapter 8 are well-researched resources that promote the analytical skills in students. The questions covered in these exercises are competent to deliver a deep understanding of all the key aspects of quadrilaterals. To prepare these exercises, visit the links of the pdf files given below.

☛ Download Class 9 Maths NCERT Solutions Chapter 8 Quadrilaterals

NCERT Class 9 Maths Chapter 8   Download PDF

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

NCERT solutions class 9 maths chapter 8 concentrates on elaborating the properties of different types of quadrilaterals, especially parallelograms that have many useful applications. These solutions are an amazing way of making learning simple and interesting. They come with a well-organized set of questions providing ample practice and opportunity to apply the acquired knowledge. To learn and practice with NCERT Solutions Class 9 Maths Chapter 8 quadrilaterals, try the exercises given below.

Chapter 8 Quadrilaterals Class 9 Maths

• Class 9 Maths Chapter 8 Ex 8.1 - 12 Questions
• Class 9 Maths Chapter 8 Ex 8.2 - 7 Questions

☛ Download Class 9 Maths Chapter 8 NCERT Book

Topics Covered: The main topics covered in class 9 maths NCERT solutions chapter 8 are the introduction of quadrilaterals, angle sum property of a quadrilateral, types of quadrilaterals, properties of a parallelogram , conditions for a quadrilateral to be a parallelogram, and the mid-point theorem.

Total Questions: Class 9 Maths Chapter 8 Quadrilaterals has a total of 19 questions, which can be subcategorized as short answer types, and moderate ones with sub-questions to facilitate the in-depth understanding of this geometric figure.

List of Formulas in NCERT Solutions Class 9 Maths Chapter 8

Students need to follow a fine preparation strategy to excel in a maths exam, like memorizing some important formulas and concepts with time-bound practice. NCERT solutions class 9 maths chapter 8 includes all the main formulas and concepts with suitable examples for students to grasp them comprehensively. Some of the most important formulas and concepts covered in these NCERT solutions for class 9 maths chapter 8 based on the angle sum property, parallelograms, and mid-point theorem are given below:

• The sum of the angles of a quadrilateral is 360 degrees.
• A quadrilateral with equal and parallel pairs of opposite sides is called a parallelogram.
• The area of a Parallelogram is equal to A = b × h.
• A diagonal of a parallelogram divides it into two congruent triangles.
• In a parallelogram, opposite sides are equal and opposite angles are equal.
• The diagonals of a parallelogram bisect each other.
• The line segment joining the mid-points of two sides of a triangle is parallel to the third side.
• The line drawn through the midpoint of one side of a triangle and parallel to another side bisects the third side.

Important Questions for Class 9 Maths NCERT Solutions Chapter 8

CBSE Important Questions for Class 9 Maths Chapter 8 Exercise 8.1

CBSE Important Questions for Class 9 Maths Chapter 8 Exercise 8.2

Video Solutions for Class 9 Maths NCERT Chapter 8

NCERT Video Solutions for Class 9 Maths Chapter 8
Video Solutions for Class 9 Maths Exercise 8.1
Video Solutions for Class 9 Maths Exercise 8.2

FAQs on NCERT Solutions Class 9 Maths Chapter 8

What is the importance of ncert solutions for class 9 maths chapter 8 quadrilaterals.

Cuemath NCERT Solutions Class 9 Maths are tailored to suit the learning potential of every child as per their grades. These solutions provide comprehension of each topic for students to gain a clear understanding of different geometric shapes. Created as per the NCERT maths textbook, these solutions are sufficient to thoroughly practice and revise the complete chapter 8 of the class 9 maths syllabus. The easy-to-understand format of these solutions is ideal for promoting the problem-solving approach required for higher-level maths studies.

What are the Important Topics Covered in NCERT Solutions Class 9 Maths Chapter 8?

The important topics included in NCERT Solutions Class 9 Maths Chapter 8 Quadrilaterals are an introduction to quadrilaterals, their types, basic properties, parallelograms, angle sum property, and midpoint theorem . NCERT solutions class 9 Maths Chapter 8 quadrilaterals nicely cover all these topics with important definitions, axioms, postulates , and examples.

Do I Need to Practice all Questions Provided in Class 9 Maths NCERT Solutions Quadrilaterals?

By regular practice of all the theorems, questions, and examples readily available in the NCERT Solutions Class 9 Maths Chapter 8, students can attain the step-wise approach to answer various types of questions asked in exams. It will also help them gain the necessary confidence required to face home exams or any competitive exams.

How Many Questions are there in NCERT Solutions Class 9 Maths Chapter 8 Quadrilaterals?

NCERT Class 9 Maths Chapter 8 quadrilaterals has a total of 19 questions in 2 exercises that adequately cover all the concepts of quadrilaterals. The problems in these exercises are compiled as per the CBSE syllabus that offers clear and precise learning of the entire topic for excellent preparation.

What are the Important Formulas in Class 9 Maths NCERT Solutions Chapter 8?

The important formulas and concepts covered in the NCERT Solutions Class 9 Maths Chapter 8 are based on the sum of quadrilaterals’ angles, types, properties, and theorems. These solutions provide a detailed knowledge of all these concepts through an engaging format with sample exercises explained in a well-organized manner. Understanding these concepts will enable students to solve problems based on them efficiently.

Why Should I Practice NCERT Solutions Class 9 Maths quadrilaterals chapter 8?

The CBSE class 9 maths exams are based on NCERT textbooks, and practicing efficiently with NCERT Solutions Class 9 Maths Chapter 8 quadrilaterals assures that none of the topics is left. Practicing NCERT solutions also help students to acquire the math skills that are beneficial for academics as well as for practical life.

Case Study Questions for Class 9 Maths

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Are you preparing for your Class 9 Maths board exams and looking for an effective study resource? Well, you’re in luck! In this article, we will provide you with a collection of Case Study Questions for Class 9 Maths specifically designed to help you excel in your exams. These questions are carefully curated to cover various mathematical concepts and problem-solving techniques. So, let’s dive in and explore these valuable resources that will enhance your preparation and boost your confidence.

Join our Telegram Channel, there you will get various e-books for CBSE 2024 Boards exams for Class 9th, 10th, 11th, and 12th.

CBSE Class 9 Maths Board Exam will have a set of questions based on case studies in the form of MCQs. The CBSE Class 9 Mathematics Question Bank on Case Studies, provided in this article, can be very helpful to understand the new format of questions. Share this link with your friends.

If you want to want to prepare all the tough, tricky & difficult questions for your upcoming exams, this is where you should hang out.  CBSE Case Study Questions for Class 9  will provide you with detailed, latest, comprehensive & confidence-inspiring solutions to the maximum number of Case Study Questions covering all the topics from your  NCERT Text Books !

Table of Contents

CBSE Class 9th – MATHS: Chapterwise Case Study Question & Solution

Case study questions are a form of examination where students are presented with real-life scenarios that require the application of mathematical concepts to arrive at a solution. These questions are designed to assess students’ problem-solving abilities, critical thinking skills, and understanding of mathematical concepts in practical contexts.

Chapterwise Case Study Questions for Class 9 Maths

Case study questions play a crucial role in the field of mathematics education. They provide students with an opportunity to apply theoretical knowledge to real-world situations, thereby enhancing their comprehension of mathematical concepts. By engaging with case study questions, students develop the ability to analyze complex problems, make connections between different mathematical concepts, and formulate effective problem-solving strategies.

• Case Study Questions for Chapter 1 Number System
• Case Study Questions for Chapter 2 Polynomials
• Case Study Questions for Chapter 3 Coordinate Geometry
• Case Study Questions for Chapter 4 Linear Equations in Two Variables
• Case Study Questions for Chapter 5 Introduction to Euclid’s Geometry
• Case Study Questions for Chapter 6 Lines and Angles
• Case Study Questions for Chapter 7 Triangles
• Case Study Questions for Chapter 8 Quadilaterals
• Case Study Questions for Chapter 9 Areas of Parallelograms and Triangles
• Case Study Questions for Chapter 10 Circles
• Case Study Questions for Chapter 11 Constructions
• Case Study Questions for Chapter 12 Heron’s Formula
• Case Study Questions for Chapter 13 Surface Area and Volumes
• Case Study Questions for Chapter 14 Statistics
• Case Study Questions for Chapter 15 Probability

The above  Case studies for Class 9 Mathematics will help you to boost your scores as Case Study questions have been coming in your examinations. These CBSE Class 9 Maths Case Studies have been developed by experienced teachers of schools.studyrate.in for benefit of Class 10 students.

• Class 9 Science Case Study Questions
• Class 9 Social Science Case Study Questions

How to Approach Case Study Questions

When tackling case study questions, it is essential to adopt a systematic approach. Here are some steps to help you approach and solve these types of questions effectively:

• Read the case study carefully: Understand the given scenario and identify the key information.
• Identify the mathematical concepts involved: Determine the relevant mathematical concepts and formulas applicable to the problem.
• Formulate a plan: Devise a plan or strategy to solve the problem based on the given information and mathematical concepts.
• Solve the problem step by step: Apply the chosen approach and perform calculations or manipulations to arrive at the solution.
• Verify and interpret the results: Ensure the solution aligns with the initial problem and interpret the findings in the context of the case study.

Tips for Solving Case Study Questions

Here are some valuable tips to help you effectively solve case study questions:

• Read the question thoroughly and underline or highlight important information.
• Break down the problem into smaller, manageable parts.
• Visualize the problem using diagrams or charts if applicable.
• Use appropriate mathematical formulas and concepts to solve the problem.
• Show all the steps of your calculations to ensure clarity.
• Check your final answer and review the solution for accuracy and relevance to the case study.

Benefits of Practicing Case Study Questions

Practicing case study questions offers several benefits that can significantly contribute to your mathematical proficiency:

• Enhances critical thinking skills
• Improves problem-solving abilities
• Deepens understanding of mathematical concepts
• Develops analytical reasoning
• Prepares you for real-life applications of mathematics
• Boosts confidence in approaching complex mathematical problems

Case study questions offer a unique opportunity to apply mathematical knowledge in practical scenarios. By practicing these questions, you can enhance your problem-solving abilities, develop a deeper understanding of mathematical concepts, and boost your confidence for the Class 9 Maths board exams. Remember to approach each question systematically, apply the relevant concepts, and review your solutions for accuracy. Access the PDF resource provided to access a wealth of case study questions and further elevate your preparation.

Q1: Can case study questions help me score better in my Class 9 Maths exams?

Yes, practicing case study questions can significantly improve your problem-solving skills and boost your performance in exams. These questions offer a practical approach to understanding mathematical concepts and their real-life applications.

Q2: Are the case study questions in the PDF resource relevant to the Class 9 Maths syllabus?

Absolutely! The PDF resource contains case study questions that align with the Class 9 Maths syllabus. They cover various topics and concepts included in the curriculum, ensuring comprehensive preparation.

Q3: Are the solutions provided for the case study questions in the PDF resource?

Yes, the PDF resource includes solutions for each case study question. You can refer to these solutions to validate your answers and gain a better understanding of the problem-solving process.

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NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Table of Contents

NCERT Solutions for Class 9 Maths Chapter 8: Quadrilaterals are an educational aid for students that help them solve and learn simple and difficult tasks. It includes a complete set of questions organized with advanced level of difficulty, which provide students ample opportunity to apply combinations and skills. Get free NCERT Solutions for Class 9 Maths devised according to the latest update on term-wise CBSE Syllabus 2024-25 . These NCERT Solutions will help the students to understand the concept of Quadrilaterals mainly basics, properties and some important theorems. These solutions can not only help students to clear their doubts but also to prepare more efficiently for the second term examination.

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NCERT Solutions Class 9 Maths Chapter 8 PDF

NCERT solutions for class 9 mathematics, Chapter 8, offer meticulously researched materials designed to enhance students’ analytical abilities. These exercises covers various facets of quadrilaterals, ensuring a comprehensive grasp of the topic. For access to these exercises, kindly follow the provided links to the respective PDF files.

• Important Questions Class 9 Maths Chapter 8 Quadrilaterals
• Class 9 Maths Chapter 8 Quadrilaterals MCQs

Download Class 9 Maths NCERT Solutions Chapter 8 Quadrilaterals

CBSE Class 9 Maths Chapter 8 explains Angle Sum Property of a Quadrilateral, Types of Quadrilaterals and Mid-Point theorem. Topics covered under this chapter help the students to understand the basics of a geometrical figure named as a quadrilateral, its properties and various important theorems. This chapter of NCERT Solutions for Class 9 Maths is extremely crucial as the formulas and theorem results are extensively used in several other maths concepts in higher grades.

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Chapter 8 Quadrilaterals is included in the second term CBSE Syllabus 2024-25 and is a part of Unit-Geometry which holds 28 marks of weightage in the term exams of CBSE Class 9 Maths. Two or three questions are asked every year in the second term examination from this chapter.

Other study resources for class 9 available at IL

NCERT Solutions For Class 9 Maths Chapter 8 Exercises

Get detailed solutions for all the questions listed under the below exercises:

Exercise 8.1 Solutions (12 Questions)

Exercise 8.2 Solutions (7 Questions) NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals NCERT Solutions for Class 9 Maths Chapter 8 is about Theorems and properties on Quadrilaterals. They are accompanied with explanatory figures and solved examples, which are explained in a comprehensive way. The main topics covered in this chapter include:

8.1	Introduction
8.2	Angle Sum Property of a Quadrilateral
8.3	Types of Quadrilateral
8.4	Properties of a Parallelogram
8.5	Another Condition for a Quadrilateral to be a Parallelogram
8.6	The Mid-point Theorem
8.7	Summary

Key Features of Using NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

• NCERT solutions have been prepared in a logical and simple language.
• Pictorial presentation of all the questions.
• Emphasizes that learning should be activity-based and knowledge-driven.
• The solutions are explained in a well-organised way.
• Step by step approach used to solve all NCERT questions.

Also Check: 5 Toughest and Easiest Chapters in CBSE Class 9 Maths

Formulas Used in NCERT Solutions Class 9 Maths Chapter 8 Quadrilaterals

To ace a math exam, students should adopt a solid study plan. This involves memorizing key formulas and concepts through regular practice within a set timeframe. NCERT Solutions for Class 9 Maths Chapter 8 provide a comprehensive understanding of essential formulas and concepts, accompanied by practical examples. Here are some crucial formulas and concepts covered:

• Angle Sum Property of a Quadrilateral: The total sum of the angles in a quadrilateral is always 360 degrees.
• Parallelograms: A quadrilateral with opposite sides equal and parallel is called a parallelogram.
• Area of a Parallelogram: The area (A) of a parallelogram is calculated using the formula A = base × height (b × h).
• Diagonals of a Parallelogram: A diagonal of a parallelogram divides it into two congruent triangles. Also, the diagonals of a parallelogram bisect each other.
• Properties of Parallelograms: In a parallelogram, opposite sides are equal in length and opposite angles are equal in measure.
• Mid-Point Theorem: The line segment joining the midpoints of two sides of a triangle is parallel to the third side. Additionally, a line drawn through the midpoint of one side of a triangle and parallel to another side bisects the third side.

These concepts are fundamental for understanding geometry and are crucial for solving problems related to quadrilaterals and triangles.

Frequently Asked Questions on NCERT Solutions for Class 9 Maths Chapter 8

What are the main topics covered in ncert solutions for class 9 maths chapter 8.

The main topics covered in NCERT Solutions for Class 9 Maths Chapter 8 are given below: 8.1 Introduction of quadrilaterals 8.2 Angle Sum Property of a Quadrilateral 8.3 Types of Quadrilaterals 8.4 Properties of a Parallelogram 8.5 Another Condition for a Quadrilateral to be a Parallelogram 8.6 The Mid-point Theorem 8.7 Summary

How many questions are there in NCERT Solutions for Class 9 Maths Chapter 8?

NCERT Solutions for Class 9 Maths Chapter 8 contains two exercises. The first exercise has 12 questions and the second exercise has 7 questions. Practising these exercises help you in scoring high in second term exams and also help to ease the subject. These solutions are explained by subject matter experts to help you in clearing all the doubts.

What is the meaning of quadrilaterals according to NCERT Solutions for Class 9 Maths Chapter 8?

According to NCERT Solutions for Class 9 Maths Chapter 8 quadrilateral is a plane figure that has four sides or edges, and also has four corners or vertices. Quadrilaterals will typically be of standard shapes with four sides like rectangle, square, trapezoid, and kite or irregular and uncharacterized shapes.

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Case Study Questions for Class 9 Maths Chapter 9 Areas of Parallelograms and Triangles

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Here we are providing case study questions for Class 9 Maths Chapter 9 Areas of Parallelograms and Triangles. Students are suggested to solve the questions by themselves first and then check the answers. This will help students to check their grasp on this particular chapter Triangles.

Case Study Questions:

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NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals are provided here. Our NCERT Maths solutions contain all the questions of the NCERT textbook that are solved and explained beautifully. Here you will get complete NCERT Solutions for Class 9 Maths Chapter 8 all exercises Exercise in one place. These solutions are prepared by the subject experts and as per the latest NCERT syllabus and guidelines. CBSE Class 9 Students who wish to score good marks in the maths exam must practice these questions regularly.

Class 9 Maths Chapter 8 Quadrilaterals NCERT Solutions

Below we have provided the solutions of each exercise of the chapter. Go through the links to access the solutions of exercises you want. You should also check out our NCERT Class 9 Solutions for other subjects to score good marks in the exams.

NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1

NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.2

NCERT Solutions for Class 9 Maths Chapter 8 – Topic Discussion

Below we have listed the topics that have been discussed in this chapter.

• The Mid-point Theorem
• Another Condition for a Quadrilateral to be a Parallelogram
• Properties of a Parallelogram
• Angle Sum Property of a Quadrilateral
• Types of Quadrilaterals

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NCERT Solutions for Class 6, 7, 8, 9, 10, 11 and 12

Important Questions for CBSE Class 9 Mathematics Quadrilaterals

October 11, 2018 by Sastry CBSE

Important Questions for CBSE Class 9 Mathematics Chapter 2 Quadrilaterals

The topics and sub-topics in Class 9 Maths Chapter 8 Quadrilaterals:

• Quadrilaterals
• Introduction
• Angle Sum Property Of A Quadrilateral
• Types Of Quadrilaterals
• Properties Of A Parallelogram
• Another Condition For A Quadrilateral To Be A Parallelogram
• The MidPoint Theorem

IMPORTANT QUESTIONS

More Resources for CBSE Class 9

NCERT Solutions

• NCERT Solutions Class 9 Maths
• NCERT Solutions Class 9 Science
• NCERT Solutions Class 9 Social Science
• NCERT Solutions Class 9 English
• NCERT Solutions Class 9 Hindi
• NCERT Solutions Class 9 Sanskrit
• NCERT Solutions Class 9 IT
• RD Sharma Class 9 Solutions

Question.6.If the diagonals of a quadrilateral bisect each other at right angles, then name the quadrilateral. Solution. Rhombus.

Question.10 If the diagonals of a parallelogram are equal, then state its name. Solution. Rectangle

Question. 15.If ABCD is a parallelogram, then what is the measure of ∠A – ∠C ? Solution. ∠ A – ∠ C = 0° [opposite angles of parallelogram are equal]

Value Based Questions (Solved)

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 Geometry
• Chapter 6 Lines and Angles
• Chapter 7 Triangles
• Chapter 8 Quadrilaterals
• Chapter 9 Areas of Parallelograms and Triangles
• Chapter 10 Circles
• Chapter 11 Constructions
• Chapter 12 Heron’s Formula
• Chapter 13 Surface Areas and Volumes
• Chapter 14 Statistics
• Chapter 15 Probability
• Class 9 Maths (Download PDF)

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Test: Quadrilaterals- Case Based Type Questions- 1 - Class 9 MCQ

10 questions mcq test - test: quadrilaterals- case based type questions- 1, harish makes a poster in the shape of a parallelogram on the topic save electricity for an inter school competition as shown in the follow figure. q. if ab = (2y – 3) and cd = 5 cm then what is the value of y.

⇒ 2y – 3 = 5

Harish makes a poster in the shape of a parallelogram on the topic SAVE ELECTRICITY for an inter school competition as shown in the follow figure. Q. If ∠B = (2y)° and ∠D = (3y – 6)°, then find the value of y.

(opposite angles of a parallelogram are equal)

⇒ 2y = 3y – 6

⇒ 2y – 3y = – 6

⇒ – y = – 6

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Harish makes a poster in the shape of a parallelogram on the topic SAVE ELECTRICITY for an inter school competition as shown in the follow figure. Q. If ∠ A = (4x + 3)° and ∠D = (5x – 3)°, then find the measure of ∠B

∠A + ∠D = 180°

(adjacent angles of a quadrilateral are equal)

(4x + 3)° + (5x – 3)° = 180°

∠D = (5x – 3)° = 97°

Thus, ∠B = 97°

Harish makes a poster in the shape of a parallelogram on the topic SAVE ELECTRICITY for an inter school competition as shown in the follow figure.

Q. If ∠A = (2x – 3)° and ∠C = (4y + 2)°, then find how x and y relate.

• A. x = 2y + 3

• D. x = y – 7

⇒ 2x – 3 = 4y + 2

⇒ 2x = 4y + 5

Q. Which mathematical concept is used here?

• A. Co-ordinate geometry
• B. Surface area and volume
• C. Properties of a parallelogram
• D. Probability

If one pair of opposite sides of a quadrilateral is equal and parallel, then the quadrilateral is a parallelogram.

During maths lab activity, teacher gives four sticks of lengths 6 cm, 6 cm, 4 cm and 4 cm to each student to make different types of quadrilateral.

She asks following questions from the students:

Q. A student formed a rectangle with these sticks. What is the length of the diagonal of the rectangle formed by the student?

6 2 + 4 2 = l 2

36 + 16 = l 2

Q. Write the name of quadrilateral that can be formed with these sticks.

• A. Kite, rectangle, rhombus
• B. Parallelogram, rectangle , trapezium
• C. Kite, rectangle, parallelogram
• D. Square, rectangle, kite

Q. How many types of quadrilaterals can be possible?

Q. Which statement is incorrect ?

• A. Opposite sides of a parallelogram are equal
• B. A kite is not a parallelogram
• C. Diagonals of a parallelogram bisect each other
• D. A trapezium is a parallelogram.

Q. A diagonal of a parallelogram divides it into two _______ triangles.

• B. Congruent
• C. Equilateral
• D. Right angled

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Important Questions for Quadrilaterals- Case Based Type Questions- 1

Quadrilaterals- case based type questions- 1 mcqs with answers, online tests for quadrilaterals- case based type questions- 1.

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NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Class 9 Maths NCERT Solutions Chapter 8 includes some very fundamental concepts that will encourage students to study the topic in a more in-depth manner. Chapter 8 consists of the basics of quadrilaterals which includes all four-sided, two-dimensional shapes.

NCERT solutions for class 9 maths chapter 8 will allow students to navigate each of the exercises present in the chapter which will undoubtedly be extremely helpful to many students referring to it.

Table of Contents

Ncert solutions for class 9 maths chapter 8 quadrilaterals exercise 8.1.

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals Exercise 8.2

NCERT Solutions for Class 9 Maths Chapter 8

This chapter consists of fundamental concepts like the properties of quadrilaterals and the mid-point theorem. These principles will be beneficial while understanding the more advanced principles that will be a part of the NCERT portions until class 10. Additionally, NCERT solutions Chapter 8 Quadrilaterals are the optimal way to enrich your revision of the content due to their detailed and intuitive explanation of the questions.

Key points of NCERT solutions for class 9 chapter 8

Parallelogram properties: Its diagonal divides it into two congruent triangles, the opposite sides are parallel and equal, opposite angles are equal, and the diagonals bisect each other

Midpoint theorem: i) The line segment joining the mid-points of two sides of a triangle will be parallel to the third side. ii) The line drawn through the midpoint of one side of a triangle, parallel to another side bisects the third side.

Additionally, it would be beneficial to revise the previous topics that have been covered thus far in your NCERT textbook such as the concepts of triangles, lines and angles and the introduction to Euclidean geometry. The solutions for NCERT class 9 maths for all of these chapters can be found on our website. Our quality solutions will guarantee that you are thorough with all prerequisites before going forward with the solutions for chapter 8 quadrilateral.

What is the Importance of NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals?

Quadrilateral solutions are an essential counterpart to your NCERT textbook in order to ensure that you get complete knowledge of the subject. NCERT Solutions for class 9 chapter 8 quadrilaterals is a detail-oriented resource that will provide an explanation for every exercise in your NCERT maths textbooks. So if you find yourself stuck on a particular problem or just want to verify the validity of your answers we would recommend that you go through the solutions. Furthermore, the explanations will help you learn actively and ensure your preparedness for exams.

What is the meaning of quadrilaterals, according to NCERT Solutions for Class 9 Maths Chapter 8?

A quadrilateral is a closed two-dimensional shape consisting of four sides, four vertices and the four internal angles of the figure giving a total of 360o. This definition includes shapes like squares, rectangles, rhombus and trapezium.

Why Should I Practise NCERT Solutions Class 9 Maths quadrilaterals Chapter 8?

You can expect at least some of the long answer questions in your exams from Class 9 NCERT Maths Chapter 8 Quadrilaterals. Considering that most of these questions are based on providing proofs and stating the correct theorems pertaining to quadrilaterals, you can score highly if you thoroughly practise NCERT solutions of class 9 maths. Furthermore, you will reduce the time you take for solving these questions which will allow you to complete your papers much more easily. Lastly, if you’re revising previous topics then solutions for class 9 maths will allow you to recall the topics better.

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Unit 11: Quadrilaterals

Kinds of quadrilaterals.

• Intro to quadrilateral (Opens a modal)
• Quadrilateral types (Opens a modal)
• Analyze quadrilaterals Get 3 of 4 questions to level up!
• Quadrilateral types Get 3 of 4 questions to level up!
• No videos or articles available in this lesson
• Polygon types Get 3 of 4 questions to level up!

Angle sum property

• Sum of interior angles of a polygon (Opens a modal)
• Sum of the exterior angles of a polygon (Opens a modal)
• Angles of a polygon Get 3 of 4 questions to level up!
• Find angles in triangles Get 5 of 7 questions to level up!
• Interior and exterior angles of a polygon Get 3 of 4 questions to level up!

Properties of a parallelogram

• Proof: Opposite sides of a parallelogram (Opens a modal)
• Proof: Opposite angles of a parallelogram (Opens a modal)
• Proof: Diagonals of a parallelogram (Opens a modal)
• Proof: Rhombus diagonals are perpendicular bisectors (Opens a modal)
• Side and angle properties of a parallelogram (level 1) Get 3 of 4 questions to level up!
• Side and angle properties of a parallelogram (level 2) Get 3 of 4 questions to level up!
• Diagonal properties of parallelogram Get 3 of 4 questions to level up!
• Properties of Parallelograms Get 5 of 7 questions to level up!

Parallel lines and triangles

• Midpoint theorem Get 5 of 7 questions to level up!

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Ncert solutions for class 9 maths chapter 8 quadrilaterals| pdf download.

• Exercise 8.1 Chapter 8 Class 9 Maths NCERT Solutions
• Exercise 8.2 Chapter 8 Class 9 Maths NCERT Solutions

NCERT Solutions for Class 9 Maths Chapters:

What are the benefits of NCERT Solutions for Chapter 8 Quadrilaterals Class 9 NCERT Solutions?

What is a rhombus, what is mid-point theorem, in a quadrilateral, ∠ a : ∠ b : ∠ c : ∠ d = 1 : 2 : 3 : 4, then find the measure of each angle of the quadrilateral., contact form.