NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals (Ex 8.1) Exercise 8.1

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The Central Board of Secondary Education is one of the most popular and largest boards that operate in India. Schools that are affiliated with CBSE are large and have schools scattered all over the country. The pattern of the CBSE curriculum and their question paper pattern is very similar, and they follow a very objective approach. The objective approach comprises a well-curated syllabus, which demands students go through their subjects and the topics encompassed in them with great depth and understand the nuances of the topics. CBSE wants its students to hone and develop their analytical as well as cognitive abilities. This enables students to look into the fundamentals of a topic and understand every aspect of the topic that is being discussed. CBSE does not encourage blindly memorizing everything that they are learning from their textbooks. CBSE wants its students to have a formidable grasp of the topics so that they can relate the ideas discussed in one chapter of one subject with another chapter of the same subject or a different subject. When students are able to connect the dots and correlate ideas from different chapters across subjects, it ensures optimum comprehension. Students must be able to answer questions that demand a very critical understanding of ideas. Students must complete the syllabus before their exams, and they should also revise the chapters. The more a student revises and revisits old chapters and subjects and solves more problems, their doubts subside, and slowly but gradually, they start to have the depth that CBSE demands from its students. Therefore, students must be adept at filtering through redundant and unnecessary pieces of information and learning only the important ideas. Students are generally very scared about the entire structure that CBSE employs for its students to follow. A CBSE classroom is very different from a traditional classroom, and therefore, the change can be daunting to students who transition to the CBSE board from other boards. Therefore, teachers and educators who have been a crucial part of Extramarks have curated and compiled resources and shared various tricks and hacks that students can employ to counter these issues.

The National Council of Educational Research and Training, NCERT, is an organization placed in India and operated centrally by the Indian Government. NCERT was set up in the year 1961 by the Indian government. The NCERT was set up so that India can finally have an autonomous organization which would legislate on matters that are related to the academic development of the country. NCERT employs individuals who are highly qualified and have relevant experience in their field. NCERT helps the Indian government to make decisions regarding complicated matters which are related to the education of the country. NCERT and CBSE have an extremely codependent relationship and function symbiotically. NCERT puts out their own sets of regulations and rules that students need to follow once they have become a part of the CBSE. NCERT implores all CBSE schools to follow all the rules and regulations that they have set in place for students. NCERT also has its own publishing house, which publishes books that adhere completely to the syllabus that professionals have curated. Therefore, when students refer to the NCERT textbooks, they get very used to the question paper pattern and help students formulate their exam strategy. CBSE requests that its students treat the NCERT textbook as their primary textbook because, according to them, this is an extremely crucial step for students who are aiming to ace their exams.

In recent years, NCERT has done unprecedented work in the academic field of the country so that the milieu improves and becomes more holistic and inclusive. There has been an incredible introduction to various provisions which were made available for students. One of them was the introduction to various inter school events where students make acquaintances and friends, and they socialize with people from different backgrounds which helps them exchange separate views on various topics which help broaden their minds. CBSE has provided various ways in which students can get themselves and their academic opinions published in esteemed journals. There are various students who are looking for a very eclectic education that focuses on diverse topics that schools all over the world teach, and therefore there are various highly efficient foreign exchange programmes for students to enlist in. CBSE and NCERT provide this amazing platform to students so that they can apply this opportunity to make a great career for their great future.

Mathematics is one of the most important subjects for every Class 9 student. CBSE employs a very objective approach when it comes to its curriculum. CBSE wants its students to understand the fundamentals of the chapter rather than blindly practising numerical problems. If a student aspires to score well in the CBSE exams, they must not only practise the sums but also have to be very observant about the problems and the problem-solving process. It is only through this particular methodology that enables students to get a formidable grasp of the topic. Almost all students in Class 9 have mathematics as a subject. Mathematics has the most interdisciplinary connections with other subjects, and before students decide after their first board exam in Class 10 which stream they will follow, they might choose among Science, Commerce and Humanities. Mathematics is also one of the few subjects that is singular and not merged with other subjects. Therefore, it is crucial for students to do well in their mathematics exams and have a good final grade. Therefore, Extramarks have come up with Class 9 Maths Chapter 8 exercise 8.1.

Chapter 8 in the Mathematics Class 9 CBSE curriculum is quadrilaterals which introduced students to various different polygons and exclusively discuss polygons with 4 sides like square, rectangle, parallelogram, kite, rhombus and trapezium. In order to help out, students and provide them with assistance, the teachers at Extramarks have published the NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 after days and months of hard work.

The NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1, provide comprehensive solutions to all the NCERT questions that are present in the textbook. The NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 contains every solution to all the unsolved questions in every chapter and its exercises. The solutions that are provided are very detailed. The solutions never fail to bring attention to the complicated steps which generally students struggle with. NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 are made and compiled by teachers at Extramarks who have years of relevant experience in teaching Class 9 CBSE students, and these teachers are highly qualified and esteemed in their individual fields. NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 is like a one-stop solution for every student who is struggling with this chapter. All the steps are very meticulously and lucidly mentioned. This enables every student, no matter what academic background they are from, to access the NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1.

NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 can be accessed by everyone on the Extramarks website. NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1, can be accessed through their new mobile application. The solutions adhere to all the rules and regulations that are put out by CBSE. The solutions are written and solved in a way that the CBSE accepts. The more a student uses the NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 the better they get at that chapter. Teachers always advise students to refer to and use the NCERT solutions. Teachers have substantiated the accuracy and legitimacy of these solutions. The NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1, chronologically follow the same sequence of all the questions that are provided in the NCERT textbook. Teachers have shared that there are various uses for these solutions. These teachers have shared their experiences and the various and efficient ways in which students have put NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 to good use.

Maths Class 9 Chapter 8 Exercise 8.1 is very important, and teachers have constantly shared their views on the importance of revising chapters that students have completed. Teachers also warn students to never leave portions off the syllabus. Mathematics is a subject that is highly connected, and to solve one problem, one often has to employ concepts from other chapters. Therefore, often students fail to attempt the full question paper because some problems are based on portions of the syllabus that they have omitted. Teachers have noted that one of the best uses of the NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 is how it helps anyone who uses them save a lot of time. The NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 is one of the best resources that were made available to students. Students can access the NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1, anytime and anywhere they want. Students are thus always under expert guidance. Teachers and students alike have shared their positive experiences with Extramark’s NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1. Teachers have vouched for the accuracy of the NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1, and they advise students to use it more.

H2 – Class 9 Maths Chapter 8 Quadrilaterals Exercise 8.1

H2 – Access NCERT solution for Class-9 Maths Chapter 8 – Quadrilateral

The NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 can be easily accessed through the Extramarks website, and they also have a mobile application through which student can access the NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 as well.

Click on the link below to be directed to the NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 where one can download the PDF for free.

H3 – EXERCISE NO: 8.1

NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 provide solutions for the 8th Chapter of the Mathematics curriculum, which is quadrilaterals.

The first chapter asks questions based on all the properties of the parallelogram and the NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 provide detailed solutions to all the questions in the exercise. The solutions are highly comprehensive and easy to understand.

H3 – NCERT Solutions for Class 9 Chapter 8 Maths Exercise 8.1

NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 are written in a very straightforward manner. These solutions are curated by highly professional people who excel in their field.

H3 – CBSE Class 9 Maths Chapter 8 Exercise 8.1 Solutions free PDF

The NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1, are available for free on the Extramarks website. The solutions can also be accessed through the mobile app. The NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1, are free for everyone to access. The solutions are written in very lucid language. The solutions provide very detailed explanations of all the steps that students generally struggle with.

Click on the link below to access the solutions to Quadrilateral Class 9 Exercise 8.1.

NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 is also available in Hindi. CBSE has a provision which allows students to appear in the CBSE Board exams in English as well as Hindi. Hindi is the national language of the country. There is a great number of students who opt to write their papers in their vernacular. Therefore, the NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 is a highly reliable and efficient resource for these students, given the dearth of proper solutions in Hindi. NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 has all correct answers and teaches students and familiarizes them with the paper pattern.

H3 – Class 9 Maths Exercise 8.1 Solutions

Click on the link below to access the NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1. The solutions are free, and the PDF is ready for download. The NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 is also available in Hindi.

H3 – Exercise 8.1 Class 9 Question 1

NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 provide the answer for the first question of the first exercise of the 8th chapter. Students are given a parallelogram, and they are asked to find the values of each of its angles with only the ratio among the angles given.

H3 – Ex 8.1 Class 9 Question 2

NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 provide an answer for the second question of the exercise. The question asks its students to prove that a parallelogram in reality is a rectangle.

H3 – Class 9 Maths Exercise 8.1 Question 3

NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1, provides an answer for the third question of the exercise. In this problem, students are asked to prove that when two separate diagonals are bisecting each other in a parallelogram, then it becomes a rhombus.

H3 – NCERT Class 9 Ex 8.1 Question 4

NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 provide the answer to the fourth question of the exercise. Here, students must prove that the diagonals of a square are equal in value and they both bisect each other.

H3 – NCERT Class 9 Ex 8.1 Question 5

NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 provide an answer to the fifth question of the exercise. Here unlike the previous problem, students have to prove the same thing for a parallelogram.

H3 – Class 9 Maths Chapter 8 Exercise 8.1 Question 6

NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1, provides an answer to the sixth question of the exercise. Here, students must learn how a parallelogram can be used to prove the deductions that are given in the question based on the data that is provided. The question has two parts and a diagram. The NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1, provide easy explanations for both questions.

H3 – Ex 8.1 Class 9 NCERT Question 7

NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 provide an answer to the seventh question of the exercise. In this sum, students are asked to prove how each diagonal of a rhombus bisects two different angles of a rhombus.

H3 – Class 9 Maths Chapter 8.1 Question 8

NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1, provides an answer to the 8th question of the exercise. In this problem, students are asked to prove two things. The first one asks to prove that a rectangle whose diagonals bisect is a square. Once that is proven, then students have to prove that the diagonals of the square bisect the other two angles as well. NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1, explains the solutions very lucidly.

H3 – NCERT Solutions Class 9 Maths ex 8.1 Question 9

NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 provide an answer to the 9th question of the exercise. In this problem, there are 5 simple proofs based on the properties of a parallelogram, and they are very well explained.

H3 – NCERT Class 9 Maths ex 8.1 Question 10

NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 provide an answer to the 10th question of the exercise. In this sum, students are asked to prove the properties of a triangle through the parallelogram.

H3 – Class 9 Maths Chapter 8.1 Question 11

NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 provide an answer to the 11th question of the exercise. In this problem, there is a figure on students who are given 6 conditions to prove.

H3 – Ex 8.1 Class 9 NCERT Question 12

NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 provide an answer to the 12th question of the exercise. This is also a question where students have to prove a few conditions based on a diagram.

H3 – Key Takeaways of NCERT Solutions Class 9 Exercise 8.1

The NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 released by Extramarks is a very useful tool for students because –

It helps students save a lot of time because students generally rely on the NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 when they have doubts, and students can access them anytime they want.
The solutions familiarize students with the question paper pattern, which is very distinct. Students who refer to the NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1, get used to the way answers must be answered in the exams to ensure maximum marks.
Regular practice with the NCERT solutions helps students revise old chapters. The more the student practise, the better they get at it. Regular use of the NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1, ensures a significant decrease in the students’ tendency to make silly errors
The NCERT solutions also help students develop great time management skills.
For students who are preparing for competitive exams that occur in India, NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 helps these students build a good foundation for the exams.

H3 – NCERT Solutions for Class 9

Click on the link below to access the NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1.

H3 – CBSE Study Materials for Class 9

There are various Extramarks materials available on the website that cater to all the students who are in Class 9. These solutions are available for every subject, and they are written in a very lucid manner. The solutions are available for every unsolved question in the NCERT textbook for every subject.

H3 – CBSE Study Materials

Extramarks have various resources available for students to refer to, and they have similar solutions like the NCERT Solutions for Class 9 Maths Chapter 8 Exercise 8.1 available for every class and all the subjects in them.

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Therefore, the solutions to Class 9 Chapter 8 Maths Exercise 8.1 is accurate as well as reliable.

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

Ans
Let four angles of quadrilateral be 3x, 5x, 9x and 13x. Then, by angle sum property in quadrilateral 3x + 5x + 9x + 13x = 360° 30x = 360°
x = 360°/30
= 12° So, the first angle of quadrilateral
= 3(12°)
= 36° The second angle of quadrilateral
= 5(12°)
= 60° The third angle of quadrilateral
= 9(12°)
= 108° The fourth angle of quadrilateral
= 13(12°)
= 156°

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

Ans

$\begin{array}{l} Given: Let ABCD be a parallelogram, in which AC = BD. \\ To prove: ABCD is a rectangle. \\ Proof: In ΔADC and ΔBCD, \\ AD = BC [Opposite sides of parallelogram are equal.] \\ DC = CD [Common] \\ A C = BD [Given] \\ ∴ ΔADC ≅ ΔBCD [By S.S.S.] \\ ∠ ADC = ∠ BCD [By C.P.C.T.] \\ but ∠ ADC + ∠ BCD = 180 ° [Co - interior angles] \\ \Rightarrow ∠ ADC = ∠ BCD = 90 ° \\ Since, one angle of parallelogram is 90°. \\ So, ABCD is a rectangle. Hence proved. \end{array}$

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

Ans

$\begin{array}{l} Given : Let ABCD be a quadrilateral, in which AO = OC, \\ BO = OD . AC and BD bisect each other at 9 0 ° . \\ To prove : ABCD is a rhombus . \\ Proof : In ΔAOB and ΔCOB \\ AO = OC [Given] \\ ∠ AOB = ∠ COB [Each 90°] \\ OB = OB [Common] \\ ∴ ΔAOB ≅ ΔCOB [By SAS] \\ \Rightarrow AB = BC . .. (i) [By C.P.C.T.] \\ In ΔBOC and ΔDOC \\ BO = OD [Given] \\ ∠ BOC = ∠ DOC [Each 90°] \\ OC = OC [Common] \\ ∴ BOC ≅ ΔDOC [By SAS] \\ \Rightarrow BC = DC . .. (ii) [By C.P.C.T.] \\ In ΔCOD and ΔDOA \\ CO = OA [Given] \\ ∠ COD = ∠ AOD [Each 90°] \\ OD = OD [Common] \\ ∴ ΔCOD ≅ ΔDOA [By SAS] \\ \Rightarrow DC = DA . .. (iii) [By C.P.C.T.] \\ From equation (i), (ii) and (iii), we get \\ AB = BC = CD = DA \\ Since, all sides of quadrilateral ABCD are equal, so \\ ABCD is a rhombus. Hence proved. \end{array}$

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

Ans

$\begin{array}{l} Given: Let ABCD be a square. \\ To prove:AC = BD, A O = OC, BO = OD and AC ⊥ BD. \\ Proof: In ΔABC and ΔDCB \\ AB = DC [Sides of square] \\ ∠ ABC = ∠ DCB [Each 90°] \\ BC = BC [Common] \\ ΔABC ≅ ΔDCB [By S.A.S.] \\ AC = BD [By C.P.C.T.] \\ In ΔAOB and ΔCOD \\ AB = CD [Sides of square] \\ ∠ AOB = ∠ COD [Vertical opposite angles] \\ ∠ BAO = ∠ DCO [Alternate interior angles] \\ ΔAOB ≅ ΔCOD [By S.A.S.] \\ AO = OC [By C.P.C.T.] \\ BO = OD [By C.P.C.T.] \\ In ΔAOB and ΔCOB \\ AB = CB [Sides of square] \\ OB = OB [Common] \\ AO = OC [Proved above] \\ ΔAOB ≅ ΔCOB [By S.A.S.] \\ ∠ AOB = ∠ COB [By C.P.C.T.] \\ and ∠ AOB + ∠ COB = 180 ° \\ \Rightarrow ∠ AOB = 90 ° \\ Thus, diagonals AC and BD are equal and bisect each \\ other at 90°. Hence proved. \end{array}$

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

Ans

$\begin{array}{l} Given : Let ABCD be a quadrilateral. AC = BD and AC ⊥ BD, \\ OA = OC and OB = OD. \\ To prove: ABCD is a square. \\ Proof:In ΔAOB and ΔCOB \\ A O = OC [Given] \\ ∠ AOB = ∠ COB [Each 90°] \\ OB = OB [Common] \\ ΔAOB ≅ ΔCOB [By S.A.S.] \\ AB = BC . .. (i) [By C.P.C.T.] \\ In ΔBOC and ΔCOD \\ BO = OD [Given] \\ ∠ BOC = ∠ DOC [Each 90°] \\ OC = OC [Common] \\ ΔBOC ≅ ΔCOD [By S.A.S.] \\ BC = CD . .. (ii) [By C.P.C.T.] \\ In ΔCOD and ΔDOA \\ CO = OA [Given] \\ ∠ DOC = ∠ DOA [Each 90°] \\ OD = OD [Common] \\ ΔCOD ≅ ΔDOA [By S.A.S.] \\ CD = DA . .. (iii) [By C.P.C.T.] \\ From equation (i), (ii) and (iii), we have \\ AB = BC = CD = DA \\ Thus, ABCD is a square.Hence proved. \end{array}$

Q.6 Diagonal AC of a parallelogram ABCD bisects ∠A . Show that
(i) it bisects ∠C and
(ii) ABCD is a rhombus.

Ans

$\begin{array}{l} Given : ABCD is a parallelogram in which diagonal AC \\ bisects ∠ A. \\ To prove : (i) it bisects ∠ C also, \\ (ii) ABCD is a rhombus \\ Proof: (i) Since, AD ∥ BC \\ so, ∠ DAC = ∠ BCA . .. (i) [Alternate interior angles] \\ and AB ∥ DC \\ so, ∠ BAC = ∠ DCA . .. (ii) [Alternate interior angles] \\ But ∠ BAC = ∠ DAC . .. (iii) [Given] \\ So, ∠ DCA = ∠ BCA [From equation (i) and (ii)] \\ \Rightarrow AC bisects ∠ C . \\ (ii) From equation (i) and equation (iii), we have \\ ∠ BAC = ∠ BCA \\ \Rightarrow BC = BA [\begin{array}{l} Sides opposite to equal angles \\ are equal. \end{array}] \\ Since, adjacent sides of parallelogram are equal, so ABCD \\ is a rhombus. Hence proved. \end{array}$

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

Ans

$\begin{array}{l} Given : ABCD is a rhombus. \\ To prove: AC bisects ∠ A as well as ∠ C. \\ BD bisects ∠ B as well as ∠ D. \\ Proof:In ΔABC, AB = BC \\ So, ∠ BAC = ∠ BCA . .. (i) [\begin{array}{l} Opposite angles of equal sides \\ are equal. \end{array}] \\ Since, AB ∥ DC \\ So, ∠ BAC = ∠ DCA . .. (ii) [Alternate interior angles.] \\ From equation (i) and equation (ii), we have \\ ∠ BCA = ∠ DCA \\ \Rightarrow diagonal AC bisects ∠ C. \\ Since, AD∥BC \\ So, ∠ DAC = ∠ BCA \dots (iii) [Alternate interior angles.] \\ From equation (i) and equation (iii), we have \\ ∠ BAC = ∠ DAC \\ \Rightarrow diagonal AC bisects ∠ A. \\ Thus, AC bisects ∠ A as well as ∠ C. \\ Similarly, BC bisects ∠ B as well as ∠ D.Hence proved. \end{array}$

Q.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.

Ans

$\begin{array}{l} Given : ABCD is a rectangle. Diagonal AC bisects ∠ A as well \\ as ∠ C . \\ To prove: (i) ABCD is a square. \\ (ii) Diagonal BD bisects ∠ B as well as ∠ D . \\ Proof: (i) Since, ∠ 1 = ∠ 2 and ∠ 3 = ∠ 4 \\ ∵ AB ∥ CD \\ ∴ ∠ 2 = ∠ 3 [Alternate interior angles] \\ \Rightarrow ∠ 1 = ∠ 3 \\ \Rightarrow CD = AD [\begin{array}{l} Opposite sides of equal angles \\ are equal. \end{array}] \end{array}$

$\begin{array}{l} Since, adjecent sides of a rectangle are equal, so ABCD is \\ a square. \\ (ii) As ABCD is a square. \\ So, AB = AD \\ \Rightarrow ∠ ADB = ∠ ABD . .. (i) [\begin{array}{l} Opposite angles of equal sides \\ are equal. \end{array}] \\ Since, AB ∥ CD \\ So, ∠ ABD = ∠ CDB . .. (ii) [Alternate interior angles] \\ \Rightarrow ∠ ADB = ∠ CDB \end{array}$ $\begin{array}{l} \end{array}$

$\begin{array}{l} \Rightarrow BD bisects ∠ D . \\ Similarly, we can prove that BD bisects ∠ B . \\ Therefore, BD bisects ∠ B as well as ∠ D . Hence proved. \end{array}$

Q.9

$\begin{array}{l} In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ . Show that : \\ (i) ΔAPD ≅ ΔCQB \\ (ii) AP = CQ \\ (iii) ΔAQB ≅ ΔCPD \\ (iv) AQ = CP \\ (v) APCQ is a parallelogram \end{array}$

Ans

$\begin{array}{l} Given : ABCD is a parallelogram and DP = BQ. \\ To prove: (i) ΔAPD ≅ ΔCQB \\ (ii) AP = CQ \\ (iii) ΔAQB ≅ ΔCPD \\ (iv) AQ = CP \\ (v) APCQ is a parallelogram \\ Proof: In ΔAPD and ΔCQB \\ AD = BC [Opposite sides of parallelogram.] \\ ∠ ADP = ∠ CBQ [Alternate r interior angles.] \\ DP = BQ [Given] \\ ∴ ΔAPD ≅ ΔCQB [By S.A.S.] \\ (ii) AP = CQ [By C.P.C.T.] \\ (iii) In ΔAQB and ΔCPD \\ AB = CD [Opposite sides of parallelogram.] \\ ∠ ABQ = ∠ CDP [Alternater interior angles.] \\ BQ = DP [Given] \\ ∴ ΔAQB ≅ ΔCPD [By S.A.S.] \\ (iv) AQ = PC [By C.P.C.T.] \\ (v) Since, AP = CQ and AQ = PC \\ So, APCQ is a parallelogram. [Opposite sides are parallel.] \\ Hence proved. \end{array}$

Q.10

$\begin{array}{l} ABCD is a parallelogram and AP and CQ are perpendiculars \\ from vertices A and C on diagonal BD (see the figure below) . \\ Show that \\ (i) ΔAPB ≅ ΔCQD \\ (ii) AP=CQ \end{array}$

Ans

$\begin{array}{l} Given:ABCD is a parallelogram and AP ⊥ BD and CQ ⊥ BD. \\ To prove : (i) ΔAPB ≅ ΔCQD \\ (ii) AP = CQ \\ Proof : (i) In ΔAPB and ΔCQD \\ ∠ ABP = ∠ CDQ [Alternate interior angles] \\ ∠ APB = ∠ CQD [Each 90°] \\ AB = CD [Opposite sides of parallelogram.] \\ ∴ ΔAPB ≅ ΔCQD [By A.A.S.] \\ (ii) AP = CQ [By C.P.C.T.] Hence proved. \end{array}$

Q.11

$\begin{array}{l} In Δ ABC and Δ DEF, AB = DE, AB | | DE, BC = EF and BC | | EF . \\ Vertices A, B and C are joined to vertices D, E and F respectively (see the figure below) . Show that \\ (i) quadrilateral ABED is a parallelogram \\ (ii) quadrilateral BEFC is a parallelogram \\ (iii) AD | | CF and AD = CF \\ (iv) quadrilateral ACFD is a parallelogram \\ (v) AC = DF \\ (vi) ΔABC ≅ ΔDEF . \end{array}$

Ans

$\begin{array}{l} Given : In ΔABC and ΔDEF, AB = DE, AB | | DE, BC = EF and \\ BC | | EF. \\ To prove: (i)quadrilateral ABED is a parallelogram \\ (ii) quadrilateral BEFC is a parallelogram \\ (iii) AD | | CF and AD = CF \\ (iv) quadrilateral A CFD is a parallelogram \\ (v) AC = DF \\ (vi) ΔABC ≅ ΔDEF . \\ Proof: (i) Since, AB = DE and AB ∥ DE \\ So, ABED is a parallelogram. \\ [One pair of opposite sides is parallel and equal.] \\ (ii) Since, BC = EF and BC | | EF \\ So, BEFC is a parallelogram. \\ [One pair of opposite sides is parallel and equal.] \\ (iii) Since, ABED is a prallelogram. \\ So, AD = BE and AD ∥ BE . .. (i) \\ [One pair of opposite sides is parallel and equal.] \\ Since, BEFC is a prallelogram. \\ So, BE = CF and BE ∥ CF . .. (ii) \\ [One pair of opposite sides is parallel and equal.] \\ From equation (i) and equation (ii), we have \\ AD = CF and AD ∥ CF \\ (iv) Since, AD = CF and AD ∥ CF \\ So, ADFC is a prallelogram. \\ [One pair of opposite sides is parallel and equal.] \\ (v) Since, ADFC is a prallelogram. \\ So, AC = DF [Opposite sides of parallelogram.] \\ (vi) In ΔABC and ΔDEF \\ AB = DE [Given] \\ BC = EF [Given] \\ AC = DF [Proved above] \\ ∴ ΔABC ≅ ΔDEF [By S . S . S .] Hence proved. \end{array}$

Q.12

$\begin{array}{l} ABCD is a trapezium in which AB | | CD and AD = BC \\ (see the figure below) . Show that \\ (i) ∠ A = ∠ B \\ (ii) ∠ C = ∠ D \\ (iii) ΔABC ≅ ΔBAD \\ (iv) diagonal AC = diagonal BD \end{array}$

Ans

$\begin{array}{l} Given : In trapezium ABCD, AB ∥ CD and AD = BC. \\ To prove: (i) ∠ A = ∠ B \\ (ii) ∠ C = ∠ D \\ (iii) ΔABC ≅ ΔBAD \\ (iv) diagona l AC = diagonal BD \\ Construction : Draw CE ∥ DA. Join AC. \\ Proof:Since, AB ∥ CD \Rightarrow AE ∥ CD \\ and CE ∥ DA \\ So, AECD is a prallelogram. \\ Then, AD = CE \\ But, AD = BC \\ \Rightarrow CE = BC \end{array}$

$\begin{array}{l} \Rightarrow ∠ CBE = ∠ CEB [\begin{array}{l} Opposite sides of equal angles are \\ equal in Δ CEB. \end{array}] \\ ∠ CBE + ∠ CBA = 180 ° . .. (i) [Linear pair of angles] \\ ∠ CEA + ∠ DAE = 180 ° [Cointerior angles] \\ ∠ CBE + ∠ DAE = 180 ° . .. (ii) [∵ ∠ CBE = ∠ CEA] \\ From equation (i) and equation (ii), we have \\ ∠ CBE + ∠ CBA = ∠ CBE + ∠ DAE \\ \Rightarrow ∠ CBA = ∠ DAE \\ \Rightarrow ∠ B = ∠ A \\ \Rightarrow ∠ A = ∠ B \\ (ii) S ince, AB ∥ DC \\ So, ∠ A + ∠ D = 180 ° \\ and ∠ B + ∠ C = 180 ° \\ \Rightarrow ∠ A + ∠ D = ∠ B + ∠ C \\ \Rightarrow ∠ D = ∠ C [∵ ∠ A = ∠ B] \\ (iii) In ΔABC and ΔBAD \\ AB = BA [Common] \\ ∠ A = ∠ B [Proved above] \\ AD = BC [Given] \\ ∴ ΔABC ≅ ΔBAD [By S.A.S.] \\ (iv) AC = BD [By C.P.C.T.] \\ Thus, diagonal AC = diagonal BD.Hence proved. \end{array}$

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NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals (Ex 8.1) Exercise 8.1

H2 – Class 9 Maths Chapter 8 Quadrilaterals Exercise 8.1