Mathematics is becoming more and more significant due to its various applications in our daily lives.. It has been applied in many fields of sciences and technology, which has accelerated its growth in various applications. As a result, professionals suggest focusing on Mathematics helps us to improvise our reasoning, analytical skills and problem solving skills. It is turning out to be a crucial subject in the primary and secondary levels. Hence, it has become part of many competitive examinations.
The main topics covered in this chapter are the basics of the parallelogram and its properties, the key concepts of rhombus and all the properties associated, the applications of the rectangle as well as square and their properties and so on..
NCERT Solutions Class 9 Mathematics Chapter 8 is based on the latest CBSE syllabus. They are effective in clearing concepts and provide authentic, reliable and to the point answers. Students are advised to follow NCERT books first and then refer to NCERT Solutions. The solutions include everything you might be searching for during last minute preparation. It saves time to a great extent. . Hence, many teachers advise students to follow NCERT Solutions for Class 9 Mathematics Chapter 8 provided by Extramarks.
Extramarks’ website is one of the fastest growing online platforms for all primary school and secondary school studies. It is trusted by lakhs of teachers and students through its relentless services. You can find all the study material right from learning to developing skills to revising on the Extramarks website. It’s a one stop solution to all your problems.
Key Topics Covered In NCERT Solutions for Class 9 Mathematics Chapter 8
An enclosed figure with four sides is called quadrilaterals. You have learnt about the different types of quadrilaterals in your lower grades. Students are familiar with basic geometrical shapes such as square, rhombus, rectangle and parallelogram. By now, you can easily differentiate between different quadrilaterals by just looking at them. But have you ever wondered how is it possible for you to do so?
The answer lies in the shape and size of the quadrilaterals.. It’s because you know that there are certain properties governing each quadrilateral which differentiate one from the other. For example, you already know from the previous classes that ‘A square has all the four sides equal’.
In Advanced Mathematics, you need to learn about its properties and find proof of the quadrilaterals. You will find everything in detail and get more practice regarding this topic by following the NCERT Solutions for Class 9 Mathematics Chapter 8 available on the Extramarks’ website.
This chapter will upgrade students’ logical and analytical skills. The way the solutions are written will help you to grasp the topics completely and will lay a strong foundation for Mathematics in higher classes.Hence, they will learn to solve any problem in a smart way. Students are advised to make full use of resources to make the most of it.
In a way Extramarks promotes learning by encouraging the students to be great learners and try to feed their insatiable curiosity through NCERT Solutions
In the previous chapter, we learnt about the triangle and its properties. In the chapter, we will cover quadrilaterals, types of quadrilaterals and their properties.
When four non-collinear points make a closed figure, then it is called a Quadrilateral. It contains four sides, four angles and four vertices.
Angle sum property of a Quadrilateral
In this section, we will learn about the angles of the Quadrilateral. You can recall from the previous classes that the sum of all angles of a Quadrilateral is 360 degrees.
To take your learning beyond books and understand better , we recommend NCERT Solutions for Class 9 Mathematics Chapter 8 available on the Extramarks’ website.
Type of Quadrilaterals
In this section, we will cover the types of the quadrilaterals covered in this chapter
When one pair of the opposite side of the quadrilateral is parallel, it is called a trapezium.
When both pairs of the opposite sides of the quadrilateral are parallel, it is called a parallelogram.
Types of parallelograms like
When one of the angles of a parallelogram is the right angle, it is called a rectangle.
When all sides are equal in the parallelogram, it is called a rhombus.
When one of the angles is a right angle, and all sides are equal in the parallelogram, it is called a square.
When two pairs of adjacent sides are equal in a quadrilateral, it is called a kite. It is not a parallelogram.
For more details, please visit our Extramarks’ website and refer to NCERT Solutions for Class 9 Mathematics Chapter 8..
Properties of a Parallelogram
An essential prerequisite to learning the properties of the parallelogram , it is necessary to brush up these theorems:
- Theorem1: A diagonal is a parallelogram that divides it into two congruent triangles.
- Theorem 2: In a parallelogram, opposite sides are equal.
- Theorem 3: If each pair of the opposite side of the quadrilateral is equal, then it is a parallelogram.
- Theorem 4: In a parallelogram, opposite angles are equal.
- Theorem 5: If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.
- Theorem 6: The diagonals of a parallelogram bisect each other.
- Theorem 7: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
Another condition for a Quadrilateral to be a Parallelogram
We have already studied the properties of a parallelogram. A quadrilateral must meet certain criteria to be a parallelogram. They can be best understood with the help of the following theorem:
- Theorem: A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.
The Mid-point Theorem
So far, you have learnt about the triangles and the quadrilaterals. Now we will turn to the midpoint of the sides of the triangle. You can understand these concepts better with the help of the theorems given below-
- Theorem1: The line segment joining the mid- point of the two sides of a triangle is parallel to the third side.
- Theorem 2: The line drawn through the midpoint point of one side of a triangle parallel to another side bisects the third side.
In this chapter, we looked at the quadrilaterals, their definitions, the sum of the angles of the quadrilateral, its properties and key theorems required to understand geometry. We can sum up the following concepts:
- Quadrilateral: When four non-collinear points make a closed figure, then it is called a quadrilateral. It contains four sides, four angles and four vertices.
- The sum of all the angles of the quadrilateral is 360 degrees.
- Types of quadrilaterals are:
- The properties of a parallelogram
This chapter has seven important theorems. .
- The conditions for a quadrilateral to be a parallelogram
- The mid-point theorem
NCERT Solutions for Class 9 Mathematics Chapter 8: Exercise & Solutions
It requires a lot of practice to be a Mathematics wizard to solve advanced levels of questions correctly. The exercises given in the NCERT textbook have different types of questions to check your understanding. You need endless practice and patience to be strong in this subject.
The complete Class 9 Mathematics Chapter 8 exercises and solutions have been included in the NCERT Solutions for Class 9 Mathematics Chapter 8, available on the Extramarks website. It has a brief overview of all the exercises along with the solutions designed as per the CBSE curriculum and guidelines.
Extramarks which tries to do away with rote learning and supplements their studies with experiential learning and other innovative educational materials. Click on the links below have to exercise specific questions and solutions for NCERT Solutions for Class 9 Mathematics Chapter 8:
- Chapter 8: Exercise 8.1 Question and answers
- Chapter 8: Exercise 8.2 Question and answers
Along with NCERT Solutions for Class 9 Mathematics Chapter 8, students can explore NCERT Solutions on our Extramarks’ website for all primary and secondary classes.
- NCERT Solutions Class 1
- NCERT Solutions Class 2
- NCERT Solutions Class 3
- NCERT Solutions Class 4
- NCERT Solutions Class 5
- NCERT Solutions Class 6
- NCERT Solutions Class 7
- NCERT Solutions Class 8
- NCERT Solutions Class 9
- NCERT solutions Class 10
- NCERT solutions Class 11
- NCERT solutions Class 12
NCERT Exemplar for Class 9 Mathematics
NCERT Exemplar helps in laying the foundation to all the basic as well as advanced concepts in such a way that the answers are self-explanatory, meaning students may not always have to depend on teachers to clarify their doubts while studying, especially during the last minute preparation making it easier to understand the concepts quickly and thoroughly. NCERT Exemplar has a repository of NCERT related questions. As a result, teachers and mentors recommend students to include NCERT Exemplar books as an integral part of their study material. It is a great guide for students to step up their preparation to get excellent results.
The book is specially designed by subject matter experts, explained in an easy to understand language, all the theorems, formulas and exercises covered in e NCERT Class 9 Mathematics textbook. You can get the NCERT Exemplar for Class 9 Mathematics from the Extramarks’ website.
Students can find questions ranging from basic to advanced level; thus, they become capable of solving all the levels of questions. After referring to Mathematics Class 9 Chapter 8 and NCERT Exemplar, students can be rest assured that nothing remains untouched and every example solution, exercise has been covered in the chapter and hence they are confident of their preparation and ace the exam with excellent results.
Key Features of NCERT Solutions for Class 9 Mathematics Chapter 8
A strong mindset always gives rise to better results. Hence, we have prepared NCERT Solutions for Class 9 Mathematics Chapter 8 in such a way that helps students to develop analytical mindset. Some of the key features are:
- The academic notes are designed in a way that helps students to manage their time efficiently
- Students learn to quickly grasp concepts, formulas, definitions, theorems, proofs , postulates and calculations. The in-text and end- text questions in the chapter , students will get enough practice to improvise their mathematical skills and enjoy solving mathematical problems.
- It enhances the confidence of the students and hence they will be able to leverage their performance.Understanding concepts would be easier if you have followed NCERT Solutions to clarify your concepts to solve long and short answer questions, MCQs, and intext and end text questions. Also, don’t forget to take notes and practise with sample papers. will make it easier to grasp topics mathematical skills
- The students will be able to interpret proofs and draw conclusions to all geometrical applications easily.
Q.1 The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.
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
So, the first angle of quadrilateral
The second angle of quadrilateral
The third angle of quadrilateral
The fourth angle of quadrilateral
Q.2 If the diagonals of a parallelogram are equal, then show that it is a rectangle.
Q.3 Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
Q.4 Show that the diagonals of a square are equal and bisect each other at right angles.
Q.5 Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
Q.6 Diagonal AC of a parallelogram ABCD bisects ∠A . Show that
(i) it bisects ∠C and
(ii) ABCD is a rhombus.
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.
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.
Q.13 ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see the figure below). AC is a diagonal. Show that :
(i) SR||AC and SR = (1/2)AC
(ii) PQ = SR
(iii) PQRS is a parallelogram.
Q.14 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.
Q.15 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.
Q.16 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.
Q.17 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.
Q.18 Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.
FAQs (Frequently Asked Questions)
Resources or other study material play an important role in every aspect of preparation. Hence, it is necessary that you have the right study material at the right time to step up your preparation. You need not get anxious or worried about it because Extramarks has the answers to all your problems. It has a repository of quality resources mentioned later in this article. We recommend students to explore these resources and pick the right material to enhance their learning experience., Extramarks, one the leading education platforms, is the right place to begin with.
Extramarks’ website is a one stop solution to all your queries and problems. . You can find NCERT textbooks, NCERT study material, NCERT revision notes, NCERT practice questions, past year papers, NCERT Solutions, and mock tests in the NCERT Solutions for Class 9 Mathematics Chapter 8.Extramarks understands how important it is to give step-by-step explanations for making the concepts easy for students and even clarify their doubts via live classes.
Mathematics is considered one of the core subjects in STEM education. Right from ancient times, people have emphasised the importance of Mathematics. As a result, it holds great value in higher studies too.
One cannot deny the fact that engineering and many scientific fields are incomplete without Mathematics. It’s one of the core subjects to focus on to enhance your career prospects as well. For instance – even the pilot cannot ensure you are flying safely if he doesn’t have the ability to think logically and better reasoning abilities. It also encourages a rational way of thinking to make life simpler and easier. No wonder Mathematics is extremely beneficial and must have a strong foundational base to enjoy this subject.