math assignment solution class 9

NCERT Solutions For Class 9 Maths – Download in PDF (2021 – 2022)

Download NCERT Solutions For Class 9 Maths in PDF

Chapter 1: number systems.

Representation of terminating/non-terminating recurring decimals on the number line (and successive magnification method). Presentation of square roots of 2, 3 and other non-rational numbers. Rationalization of real numbers, laws of integral powers and rational exponents with positive real bases in Number Systems .

• Study Online (English/Hindi)
• Exercise 1.1
• Exercise 1.2
• Exercise 1.3
• Exercise 1.4
• Exercise 1.5
• Exercise 1.6
• Assignment based on Chapter 1 in English
• NCERT Books for Class 9
• प्रश्नावली 1.1
• प्रश्नावली 1.2
• प्रश्नावली 1.3
• प्रश्नावली 1.4
• प्रश्नावली 1.5
• प्रश्नावली 1.6
• Assignment based on Chapter 1 in Hindi

Chapter 2: Polynomials

Examples and definition of a polynomial, coefficient, degrees, zeroes and terms of a polynomial. Constant, linear, quadratic and cubic polynomials, monomials, binomials, trinomials. Factors and multiples, Remainder and factor theorems, factorization of a polynomials using factor theorem.

• Exercise 2.1
• Exercise 2.2
• Exercise 2.3
• Exercise 2.4
• Exercise 2.5
• Assignment based on Chapter 2 in English
• प्रश्नावली 2.1
• प्रश्नावली 2.2
• प्रश्नावली 2.3
• प्रश्नावली 2.4
• प्रश्नावली 2.5
• Assignment based on Chapter 2 in Hindi

Chapter 3: Coordinate Geometry

In this chapter  Coordinate Geometry we will study the concept of cartesian plane, coordinates of a point in this xy – plane, name, terms, notations and other terms associated with the coordinate plane. Abscissa and ordinate of a points. Plotting a point in xy – plane and naming it.

• Exercise 3.1
• Exercise 3.2
• Exercise 3.3
• Assignment based on Chapter 3 in English
• प्रश्नावली 3.1
• प्रश्नावली 3.2
• प्रश्नावली 3.3
• Assignment based on Chapter 3 in Hindi

Chapter 4: Linear Equations in Two Variables

This chapter provides the introduction to the equation in two variables of the type ax + by + c = 0. Proving a linear equation has infinite number of solutions. Plotting a linear equation on graph and justification of any point on line. Problems based on Linear Equations in Two Variables in daily life.

• Exercise 4.1
• Exercise 4.2
• Exercise 4.3
• Exercise 4.4
• Assignment based on Chapter 4 in English
• NCERT books
• प्रश्नावली 4.1
• प्रश्नावली 4.2
• प्रश्नावली 4.3
• प्रश्नावली 4.4
• Assignment based on Chapter 4 in Hindi

Chapter 5: Introduction to Euclid’s Geometry

Euclid’s geometry and history of Indian geometry. Introduction to Euclid’s Geometry provides a way of defining the common geometrical shapes and terms. Euclid’s five postulates and redefining (equivalent versions) the fifth postulates.  Relationship between axiom, postulates and theorems.

• Exercise 5.1
• Exercise 5.2
• Assignment based on Chapter 5 in English
• प्रश्नावली 5.1
• प्रश्नावली 5.2
• Assignment based on Chapter 5 in Hindi

Chapter 6: Lines and Angles

There are two theorems in Lines and Angles chapter which may be asked for proof. First is “If two lines intersect, vertically opposite angles are equal” and second is “The sum of the angles of a triangle is 180.”. Rest theorems are given for motivations and questions will be asked on the basis of all these theorems (Motivate/Prove).

• Exercise 6.1
• Exercise 6.2
• Exercise 6.3
• Assignment based on Chapter 6 in English
• प्रश्नावली 6.1
• प्रश्नावली 6.2
• प्रश्नावली 6.3
• Assignment based on Chapter 6 in Hindi

Chapter 7: Triangles

In this chapter Triangles , two theorems (“Two triangles are congruent if any two angles and the included side of one triangle is equal to any two angles and the included side of the other triangle (ASA Congruence)” and “The angles opposite to equal sides of a triangle are equal.”) are given for proof and the rest will be asked in the form of applications/problems.

• Exercise 7.1
• Exercise 7.2
• Exercise 7.3
• Exercise 7.4
• Exercise 7.5
• Assignment based on Chapter 7 in English
• प्रश्नावली 7.1
• प्रश्नावली 7.2
• प्रश्नावली 7.3
• प्रश्नावली 7.4
• प्रश्नावली 7.5
• Assignment based on Chapter 7 in Hindi

Chapter 8: Quadrilaterals

The chapter Quadrilaterals contains only one theorem( “The diagonal divides a parallelogram into two congruent triangles”) for proof. Other will be asked in the form of application and conceptual questions. Questions are on the basis of properties of quadrilaterals and combinations of it with triangles.

• Exercise 8.1
• Exercise 8.2
• Assignment based on Chapter 8 in English
• प्रश्नावली 8.1
• प्रश्नावली 8.2
• Assignment based on Chapter 8 in Hindi

Chapter 9: Areas of Parallelograms and Triangles

The combinations of areas of parallelograms and triangles are given to prove in most of the question. Proof and questions based on “Parallelograms on the same base and between the same parallels have the same area” will be asked. Example of median may be used as theorem in most of the questions.

• Exercise 9.1
• Exercise 9.2
• Exercise 9.3
• Exercise 9.4
• Assignment based on Chapter 9 in English
• प्रश्नावली 9.1
• प्रश्नावली 9.2
• प्रश्नावली 9.3
• प्रश्नावली 9.4
• Assignment based on Chapter 9 in Hindi

Chapter 10: Circles

“Equal chords of a circle subtend equal angles at the center” and “The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle” are given for proof in Circles . The other theorems are important for solving questions based on triangle, quadrilateral and circles.

• Exercise 10.1
• Exercise 10.2
• Exercise 10.3
• Exercise 10.4
• Exercise 10.5
• Exercise 10.6
• Assignment based on Chapter 10 in English
• प्रश्नावली 10.1
• प्रश्नावली 10.2
• प्रश्नावली 10.3
• प्रश्नावली 10.4
• प्रश्नावली 10.5
• प्रश्नावली 10.6
• Assignment based on Chapter 10 in Hindi

Chapter 11: Constructions

There are two categories of constructions . One is Construction of bisectors of line segments and angles of measure 60, 90, 45 etc. The other is construction of a triangle given its base, sum/difference of the other two sides and one base angle & given perimeter and base angles.

• Exercise 11.1
• Exercise 11.2
• Assignment based on Chapter 11 in English
• प्रश्नावली 11.1
• प्रश्नावली 11.2
• Assignment based on Chapter 11 in Hindi

Chapter 12: Heron’s Formula

Use of Heron’s  formula to find the area of triangles, quadrilaterals (dividing it into two triangles) and other polygons. Knowledge of formulae of plane figures will also help in doing questions.

• Exercise 12.1
• Exercise 12.2
• Assignment based on Chapter 12 in English
• प्रश्नावली 12.1
• प्रश्नावली 12.2
• Assignment based on Chapter 12 in Hindi

Chapter 13: Surface Areas and Volumes

Students are familiar with surface areas and volumes as they have already studies mensuration in earlier classes. This chapter also contains problems based on surface areas and volumes of cube, cuboids, cylinders, cones, spheres and hemispheres. Conversion of one figure into the other comparing volumes is also given as an application of mensuration.

• English Medium
• Hindi Medium
• Assignment based on Chapter 13 in English
• Assignment based on Chapter 13 in Hindi

Chapter 14: Statistics

Introduction to statistics includes the presentation of data collected in raw form. Presentation of  data in tabular form by grouping them in proper intervals, drawing a bar graph, histogram or polygon. Finding the measure of central tendency mean, mode and median of raw data.

• Assignment based on Chapter 14 in English
• Assignment based on Chapter 14 in Hindi

Chapter 15: Probability

Probability based on observation or frequency approach. Questions based on real life or day to day incidents. Toss of a coin, throwing a dice, based on deck of cards, etc. questions based on simple events.

• Exercise 15.1
• प्रश्नावली 15.1
• Assignment based on Chapter 15 in English
• Assignment based on Chapter 15 in Hindi

Related Pages:

• Solutions For 9th Science
• Solutions For 9th Social Science
• Solutions For 9th Hindi
• Solutions For 9th English
• CBSE Syllabus 2021 – 2022
• NCERT Solutions for Class 6
• NCERT Solutions for Class 7
• NCERT Solutions For Class 8
• NCERT Solutions for Class 9
• NCERT Solution for Class 10
• NCERT Solutions for Class 11
• NCERT Solutions for Class 12

Class 10 Maths Solutions Chapter 1 Class 10 Maths Solutions Chapter 2 Class 10 Maths Solutions Chapter 3 Class 10 Maths Solutions Chapter 4 Class 10 Maths Solutions Chapter 5

Class 10 Maths Solutions Chapter 6 Class 10 Maths Solutions Chapter 7 Class 10 Maths Solutions Chapter 8 Class 10 Maths Solutions Chapter 9 Class 10 Maths Solutions Chapter 10

Class 10 Maths Solutions Chapter 11 Class 10 Maths Solutions Chapter 12 Class 10 Maths Solutions Chapter 13 Class 10 Maths Solutions Chapter 14 Class 10 Maths Solutions Chapter 15

Class 9 Maths Solutions Chapter 1 Class 9 Maths Solutions Chapter 2 Class 9 Maths Solutions Chapter 3 Class 9 Maths Solutions Chapter 4 Class 9 Maths Solutions Chapter 5

Class 9 Maths Solutions Chapter 6 Class 9 Maths Solutions Chapter 7 Class 9 Maths Solutions Chapter 8 Class 9 Maths Solutions Chapter 9 Class 9 Maths Solutions Chapter 10

Class 9 Maths Solutions Chapter 11 Class 9 Maths Solutions Chapter 12 Class 9 Maths Solutions Chapter 13 Class 9 Maths Solutions Chapter 14 Class 9 Maths Solutions Chapter 15

NCERT Solutions for Class 9 Maths

NCERT solutions for class 9 maths comprise the analysis and explanation of all the problems covered by the NCERT textbook for 9th grade. The solutions are provided in a detailed manner to ensure that students can understand concepts as they go through them. These class 9 maths NCERT solutions are prepared by some of the most accomplished experts in the field of Mathematics.

Class 9th Maths NCERT Solutions are extremely helpful for students preparing for CBSE exams and state or national competitive tests such as Olympiads. The CBSE examinations follow the structure provided by the NCERT textbooks. Additionally, CBSE NCERT solutions for class 9 maths help in getting students ready to face class 10th and the corresponding board exams.

NCERT Maths Class 9 Solutions PDF Download

Practicing these class 9 maths NCERT solutions can help students to increase their theoretical skills and practical knowledge. The solutions to all NCERT class 9 maths sums are available for free pdf download. You can find tips and tricks included with the solutions. The links to these pdf solutions are given below.

Class 9th Maths NCERT Solutions Chapter 1 to 15

As a student, you must indulge in constant revision and practice all the solutions. Thus, revisiting the above-mentioned links periodically is a must. Start by attempting the questions in the NCERT book and then have a look at these class 9th maths NCERT solutions. You can download the NCERT textbook given below and begin your learning journey.

☛ Download Class 9 Maths NCERT Book

NCERT Solutions for Class 9 Maths Chapter 1

NCERT solutions for class 9 maths chapter 1 number systems help students amass the tools required to identify rational and irrational numbers , locate real numbers on a number line , and convert a real number to its decimal form. The solutions to this chapter also give students a good understanding of simplifying expressions using certain identities and laws of exponents .

Important Formulas:

• √(ab) = √a √b
• √(a/b) = √a/√b
• (√a + √b)(√a - √b) = a - b
• (√a + √b)2 = a + b + 2√(ab)

Class 9 Maths Chapter 1 Real Numbers

Topics Covered: NCERT solutions for class 9 maths cover questions based on rationalizing the denominator , the rules of exponentiation, identities applied to positive real numbers, expanding a real number into its decimal form, and vice versa.

Total Questions: Chapter 1 has a total of 27 questions, out of which 5 are easy, 7 are moderate and 10 are difficult.

NCERT Solutions for Class 9 Maths Chapter 2

NCERT solutions for class 9 maths chapter 2 polynomials help students form the basics of polynomials that will be used in higher classes. Students learn about the types of polynomials in one variable, degree and zeros of a polynomial , the remainder theorem , factorization of polynomials , and the application of algebraic identities.

• (x + y) 2 = x 2 + 2xy + y 2
• x 2 - y 2 = (x + y) (x - y)
• (x + a) (x + b) = x 2 + (a + b)x + ab
• (x - y) 2 = x 2 - 2xy + y 2
• (x + y + z) 2 = x 2 + y 2 + z 2 + 2xy + 2yz + 2zx

Class 9 Maths Chapter 2 Polynomials

Topics Covered: Factorization of different polynomials using the splitting method and factor theorem , finding the remainder of a polynomial, applying algebraic identities to simplify sums, and determining the zeros of a polynomial.

Total Questions: Class 9 maths chapter 2 polynomials have 33 questions. 10 are of a moderate level, 14 are easy and 9 are challenging sums.

NCERT Solutions for Class 9 Maths Chapter 3

NCERT solutions for class 9 maths chapter 3 coordinate geometry help students to get a grasp on how to describe the position of an object using coordinates and explanation of the cartesian system . It acts as a foundation for further chapters that require students to use their knowledge to solve geometrical problems.

Class 9 Maths Chapter 3 Coordinate Geometry

Topics Covered: Determining the x and y coordinates of a point in a cartesian plane, identifying quadrants, locating an object, and expressing it in the form of coordinates are the topics under class 9 maths NCERT solutions.

Total Questions: There are 6 questions in class 9 maths chapter 3 coordinate geometry. 3 are difficult, 2 are medium level and 1 is an easy question.

Class 9 Maths Chapter 3 Summary:

Class 9 maths NCERT solutions chapter 3 gives a brief overview of the definition of the origin, how to use the horizontal x-axis and vertical y-axis to pinpoint the position of an object. It also helps students determine what quadrant does a point lies in depending upon the signs of the coordinates.

NCERT Solutions for Class 9 Maths Chapter 4

NCERT solutions for class 9 maths chapter 4 linear equations in two variables introduce students to a linear equation in two variables by helping them understand the concepts behind it. The solutions include solving linear equations , graphical representation methods, and the depiction of parallel lines.

Linear equations are one of the most important chapters covered in class 9 maths NCERT solutions . The following are a few tips that can help you power through these solutions quickly and efficiently.

• Build an understanding of the concepts
• Revision of the topic at regular intervals
• Understanding the general form of a line

Class 9 Maths Chapter 4 Linear Equations in Two Variables

Topics Covered: Finding solutions to a linear equation by substituting values, representing a linear equation in two variables graphically, and how to write the equation of lines parallel to the x-axis.

Total Questions: Chapter 4 linear equations in two variables have a total of 16 questions. 8 require students to carefully think and attempt the problem, 5 are moderate and 4 are easy sums.

NCERT Solutions Class 9 Maths Chapter 5

NCERT solutions class 9 maths chapter 5 introduction to Euclid’s geometry gives students an insight into the definitions of points, lines, surfaces, and so on. Students also get an idea of the 5 postulates given by Euclid that are used to solve geometry problems.

Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry

Topics Covered: Euclid’s five postulates dealing with line segments, radii, perpendicular lines , and the variation of the fifth postulate are the topics covered under NCERT solutions class 9 maths.

Total Questions: Chapter 5 introduction to Euclid’s Geometry has 9 questions, 4 are simple, 3 are intermediate and 2 are tough sums.

The chapter deals with five postulates given by Euclid regarding a straight line, a terminating line, a circle, and its radius, as well as right angles . Students also learn about the two equivalent versions of Euclid’s fifth postulate that led to the discovery of other theorems.

Class 9 Maths NCERT Solutions Chapter 6

Class 9 maths NCERT solutions chapter 6 lines and angles give students an understanding of the basic definitions of various terms that are used in geometry . Students learn about the different types of angles, pairs of angles, transversals, parallel lines , angle sum property of triangles, and axioms associated with these concepts.

Class 9 Maths NCERT Solutions Chapter 6 Preparation Tips

There are a number of properties taught in this chapter that students must learn. The following points can help in streamlining this process.

• Practice several questions on properties of angles that get formed when a transversal intersects 2 or 3 parallel lines.
• Understand the reasoning behind theorems such as the angle sum property of a triangle.
• Make a note of the relationship between pairs of angles.

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Class 9 Maths Chapter 6 Lines and Angles

Topics Covered: Adjacent angles, vertically opposite angles, linear pairs of angles, alternate angles , corresponding angles , parallel lines, and transversal are topics falling under Class 9 maths NCERT solutions. In addition to this, problems based on the angle sum property of a triangle are also included in these solutions.

Total Questions: There are a total of 18 questions in class 9 maths chapter 6 lines and angles. 10 are very easy property-based questions, 5 require some thinking on a student’s part and 3 are difficult sums.

NCERT Solutions for Class 9 Maths Chapter 7

NCERT solutions for class 9 maths chapter 7 triangles are a fantastic way for students to learn about the congruence of triangles using the SAS, SSS, ASA, RHS criteria. Several properties of triangles along with concepts on the inequality of triangles also fall under the scope of this chapter.

Class 9 Maths Chapter 7 Triangles

Topics Covered: Questions based on applying properties of triangles, proving the congruence of two triangles using different methods, and theorems on the inequality of triangles are the topics covered.

Total Questions: Class 9 maths chapter 7 has 31 questions. 10 are simple questions based on applying the properties of triangles, 9 are complex sums and 12 are moderate problems.

Chapter Summary:

In this chapter, students learn about the SSS, SAS, ASA, and RHS congruence criteria. Students can get a good grasp on properties of triangles such as sides opposite to equal angles are equivalent, the sum of any two sides of a triangle is greater than the third side, and so on. Thus, after studying this chapter students can master the basic concepts of a triangle.

NCERT Solutions for Class 9 Maths Chapter 8

NCERT solutions for class 9 maths chapter 8 quadrilaterals comprise questions that teach students about the different types of quadrilaterals such as squares , rhombuses , rectangles , parallelograms, and their properties.

Class 9 Maths Chapter 8 Quadrilaterals

Topics Covered: Angle sum property of quadrilaterals, properties of a parallelogram , square, and rhombus that depend upon different factors such as length of diagonals, the measure of sides, etc. It also includes the mid-point theorem .

Total Questions: 19 sums - 8 are simple problems, 6 are of an intermediate level and 5 are challenging questions.

Class 9 Maths NCERT Solutions Chapter 9

Class 9 maths NCERT solutions chapter 9 areas of parallelograms and triangles include various theorems that help to simplify finding the area of complex parallelograms as well as triangles, properties to find the area of triangles divided by a median, and area of planar regions.

• Area of a triangle = ½ × base × height
• Area of a parallelogram = base × height

Class 9 Maths Chapter 9 Areas of Parallelograms and Triangles

Topics Covered: Topics covered incorporate questions on finding the area of two or more triangles and parallelograms with the same base between the same parallels, determining the area of triangles that are split by a median as well as the area of congruent figures.

Total Questions: Chapter 9 areas of parallelograms and triangles have a total of 31 questions. 8 questions belong to an optional exercise with a high difficulty level, 14 are intermediate and 9 are simple sums.

NCERT Solutions for Class 9 Maths Chapter 10

NCERT solutions for class 9 maths chapter 10 circles build a foundation for students by introducing them to the various terms related to circles. Students get the opportunity to build their knowledge base on properties of chords, the angle subtended by chord and arc, perpendicular to a point, the distance of a chord from the center as well as cyclic quadrilaterals .

Preparation Tips:

NCERT solutions for class 9 maths chapter 10 are purely based on the properties of various elements of a circle. To memorize these theorems you can follow the topics listed below.

• Make a note of all the properties so that you can skim through them when required.
• Solve all NCERT textbook questions.
• Regularly revise the solutions so that you can recall them when necessary.

Class 9 Maths Chapter 10 Circles

Topics Covered: NCERT solutions class 9 maths encompass topics such as angles subtended by a chord at a point, perpendiculars dropped from the center to a chord, and how to calculate the distance of equal chords from the center. Other discussions include angles subtended by an arc of the circle, cyclic quadrilaterals, and their attributes.

Total Questions: There are a total of 35 questions in this chapter out of which 15 are relatively simpler, 10 are complex and 10 are of moderate complexity.

NCERT Solutions for Class 9 Maths Chapter 11

NCERT solutions for class 9 maths chapter 11 constructions will teach students how to precisely use geometrical instruments for basic constructions. Students learn to construct different angles , perpendicular bisectors , and triangles with a few given restraints.

Class 9 Maths Chapter 11 Constructions

Topics Covered: Constructing an angle from a ray, bisector of a given angle, and techniques to construct triangles when certain constraints such as the perimeter are mentioned, are the topics explained in this chapter.

Total Questions: Chapter 11 constructions have a total of 10 questions. 3 can be simply solved, 5 are moderate questions and 2 are difficult.

After studying NCERT solutions class 9 maths chapter 11 students will be adept at using a ruler and compass to bisect a given angle, construct angles of discrete values, draw the perpendicular bisector of a line segment , and building triangles.

NCERT Solutions for Class 9 Maths Chapter 12

NCERT solutions for class 9 maths chapter 12 heron’s formula is a formula given by the Hero of Alexandria to calculate the area of a triangle when the lengths of all three sides are known. In this chapter, students will learn more about the application of this formula to find the area of triangles and quadrilaterals .

• Semi Perimeter s = (a + b + c)/2
• Area of a Triangle = √[s (s - a) (s - b) (s - c)]

Class 9 Maths Chapter 12 Heron’s Formula

Topics Covered: NCERT Solutions class 9 maths chapter 12 cover the complete scope of Heron’s formula and how to apply it to questions for determining the area of a triangle. Additionally, this formula is also used for computing the area of a quadrilateral .

Total Questions: Chapter 12 heron’s formula consists of 15 well-curated questions out of which 5 are simple, 3 and medium level and 7 are difficult problems.

Class 9 Maths NCERT Solutions Chapter 13

Class 9 maths NCERT solutions chapter 13 surface areas and volumes introduce students to the concepts required to find the total surface areas, lateral or curved surface areas, and volumes of different shapes. The concepts and formulas in this chapter are very important as they form the basis of topics covered in higher education.

• TSA of a Cuboid = 2(lb + bh + hl)
• TSA of a Cube = 6a 2
• TSA of a Cylinder = 2πr(r + h)
• TSA of a Cone = πr(l + r)
• TSA of a Sphere = 4πr 2
• TSA of a Hemisphere = 3πr 2

Class 9 Maths Chapter 13 Surface Areas and Volumes

Topics Covered: Finding the TSA, LSA/CSA, along with the volume of shapes such as cubes, cuboids, right circular cylinder , right circular cone , sphere, and hemisphere.

Total Questions: Chapter 13 surface areas and volumes have 75 questions. 25 are easy formula-based sum, 31 are medium level and 10 are difficult problems that challenge a student’s thinking abilities.

Class 9 Maths NCERT Solutions Chapter 14

Class 9 maths NCERT solutions chapter 14 statistics provide an in-depth analysis of problems that are based on the collection and presentation of data for analytical purposes, graphical representation of data using different types of graphs, and measures of central tendency . This chapter enables students to develop their calculation speed and visualization skills.

Class 9 Maths Chapter 14 Statistics

Topics Covered: Representation of data in the form of frequency tables and drawing inferences from them, plotting data points using bar graphs , histograms , and frequency polygons, finding the mean , median , and mode of the given data .

Total Questions: Chapter 14 statistics have 26 problems that can be split into 8 long answer-type questions, 11 moderate, and 7 simple sums.

Once students finish practicing class 9 maths NCERT solutions chapter 14 they will have a good understanding of the purpose of well-organized data, the three measures of central tendency for ungrouped data namely mean, median, and mode. Students can effectively apply these formulas to all levels of problems after studying this topic.

NCERT Solutions Class 9 Maths Chapter 15

NCERT solutions class 9 maths chapter 15 probability help students deal with daily-life problems that require the use of probability concepts. This chapter helps students explore more about empirical probability, the definition of an event, and how to calculate the probability of different events.

NCERT Maths Class 9 Chapter 15 Study Tips

NCERT solutions class 9 maths chapter 6 probability is a widely used topic that people require to perform their day-to-day operations. Given below are some study tips that can help in comprehending this chapter in a better manner.

• Apply the concept to simple daily life questions.
• Practice the solutions regularly to get the best results.
• Read the question carefully and break it down to get a better idea of what has to be done.

NCERT Maths Class 9 Chapter 15 Probability

Topics Covered: Finding the probability of the occurrence of heads or tails when a coin is flipped, tossing a die, and drawing conclusions from the result obtained are the topics covered in NCERT solutions class 9 maths.

Total Questions: Chapter 15 probability has 13 questions. 4 are short answer problems, 3 are difficult and 6 are moderate sums.

Class 9 Maths NCERT Solutions Important Formulas

Class 9 maths covers several topics such as polynomials, number systems, geometry, probability, etc. The formulas and concepts behind these topics are required to understand higher grade mathematics. Formulas are required to reduce the time taken to solve questions and increase accuracy. Given below is an overview of the important maths formulas encompassed in the NCERT solutions for class 9 maths.

• Area of a triangle = ½ * b × h
• Perimeter of a triangle = a + b + c
• Area of parallelogram = b × h
• Perimeter of a parallelogram = 2(a + b)

Surface Area and volume:

• TSA of a cuboid = 2(lb + bh +lh)
• Volume of a cuboid = l×b×h
• TSA of a cube = 6a 2
• Volume of a cube = a 3
• (a + b) 2 = a 2 + b 2 + 2ab
• (a - b) 2 = a 2 + b 2 - 2ab
• (a + b) (a - b) = a 2 - b 2

Importance of NCERT Solutions for Class 9 Maths

Class 9 Maths NCERT Solutions are extremely helpful for CBSE students as they give a means for efficient time management and improving auxiliary abilities. The following are the advantages of these solutions.

• Practical Applications: Many of the sums provided by the NCERT textbook are similar to real-life examples, especially when word problems are considered. By following the class 9 maths NCERT solutions students can come up with precise solutions that can also be applied to daily life problems.
• Practice: To solve the toughest sums within a matter of minutes you need due diligence and regular practice. These NCERT solutions for class 9 maths are very helpful for this purpose as they give students a detailed overview of how to solve different types of sums. They can also help a student to complete the syllabus well within the given time frame.
• Regular Revision: On examination days, students do not have the time to solve each and every question. Thus, these NCERT solutions class 9 maths help students with a quick revision of the entire syllabus. This also helps to free up time so that students can focus on ensuring that their concept formation is correct and can be accurately applied to sums.

FAQs on NCERT Solutions for Class 9 Maths

Why are ncert solutions for class 9 maths important.

NCERT solutions for class 9 maths are vital for students as they provide reliable solutions that students can use to cross-check their answers. These class 9 maths NCERT solutions also guide students on how to appear for an examination so that students can write their answers without missing any steps and get the best possible score.

Where Can I Get Chapter-wise Solutions for NCERT Class 9 Maths?

The chapter-wise solutions for NCERT class 9 maths can be accessed by the links mentioned above. They are further segregated into exercise-wise pdfs that are available for free download. These pdfs are also compatible with mobile screens and are in a scrollable format for easy use.

How CBSE Students can utilize Class 9 Maths NCERT Solutions?

To effectively utilize class 9 maths NCERT solutions students need to ensure that they have an understanding of the concepts used to solve the problems. Thus, when they start consulting these solutions students will comprehend them in a better manner and will be able to retain what they learned more efficiently.

Do I Need to Make Notes while Referring to NCERT Solutions Class 9 Maths?

Notes act as a simplified guide for students when they are trying to build an understanding of the subject or if they are revising that question. It is a good way of retaining and recalling concepts when required. Thus, it is crucial that students make notes of the NCERT solutions class 9 maths to expedite their learning process.

What are the Best Ways to Learn Concepts Covered in NCERT Solutions Class 9th Maths?

The most impactful way to learn concepts covered in NCERT solutions class 9th maths is by constantly practicing these solutions. If a student finds some concept difficult and cannot proceed further while attempting a problem he needs to immediately clarify his doubts so that there is no gap in learning.

Can We Use Other Methods to Solve Problems Apart From Those Mentioned in NCERT Solutions for Class 9 Maths?

You may not want to use the method given in the NCERT solutions for class 9 maths however, the approach used and the answer obtained need to be universally recognized. Having said that you cannot use the tricks that are useful in solving competitive exam problems as these are not standardized techniques.

Why Choose Cuemath for Class 9 Maths NCERT Solutions?

The best website for class 9 maths NCERT solutions is Cuemath. These solutions are prepared by IITians and certified math experts. Thus, they are reliable, precise and give the best techniques to solving any problem. They also include some tips to enable students to solve sums with speed and accuracy.

How Many Chapters are there in NCERT Solutions for Class 9 Maths?

There are 15 chapters in class 9 maths NCERT solutions. The problems covered by these are fantastic to improve a student’s understanding of different concepts. The NCERT solutions give an in-depth analysis of all exercise problems.

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

All questions that are included in the CBSE home examinations are based on the NCERT class 9 maths. Thus, in order to get good marks, it is necessary to be well-versed with all the questions covered in class 9 maths NCERT solutions that can only be achieved through regular practice and due diligence.

Are NCERT Solutions for Class 9 Maths Sufficient for CBSE Exams?

The NCERT solutions for class 9 maths are sufficient for all CBSE exams. As the school teachers set the paper they include questions from the NCERT textbook. The language or values given in the problem may differ however, the approach to solving it is more or less the same. As long as a student has a robust understanding of all chapters, he can successfully clear the examination.

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

All the topics such as construction, circles , surface area , volume , statistics, etc. that are covered in the NCERT solutions for class 9 maths are required for class 10th. Thus, skipping a topic is not advisable. In addition to this, all topics within the scope of mathematics are interlinked making it necessary to give equal attention without leaving out anything.

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

The questions in NCERT solutions class 9 maths are mostly based on the application of theorems and axioms to prove a certain point. Thus, as there are a fewer number of formulas it is not only easy but necessary to memorize them all.

What are the Benefits of Class 9 Maths NCERT Solutions for CBSE Students?

The class 9 maths NCERT solutions help students to finish their syllabus on time giving them enough room for revision. They also provide a quick overview of all the problems that students can use as a reference while skimming through the chapter.

Why Should I Practice NCERT Solutions for Class 9 Maths Regularly?

The practice of NCERT solutions for class 9 maths will help a student achieve the goal of perfection. It helps students get comfortable with the covered concepts. Thus, during an examination students can keep their cool and breeze through the paper without getting flustered.

How is Class 9 Maths NCERT Solutions Promoting Problem Solving in Students?

NCERT solutions for class 9 maths start by explaining the questions and then breaking the problem down to elucidate it in a better way. It teaches students to think of the why and how behind a sum. These solutions encourage students to use a similar approach enabling them to harness their problem-solving skills and computational mindset.

NCERT Solutions for Class 9 Maths Free PDF Download

Ncert solutions for class 9 maths.

NCERT solutions for class 9 Maths is available to download for free from the links below. NCERT solutions are really helpful when it comes to a complicated subject like Mathematics. NCERT solutions for class 9 Maths will help you to understand and solve complex problems easily. It is time-saving and best study materials to follow for all the boards and especially CBSE boards.

NCERT solutions also help in cracking many competitive entrance exams like JEE Main, JEE Advanced, NEET, etc.

Class 9 Maths NCERT Solutions are available chapterwise on the app Toppr. Toppr app provides you a lot of free study materials, 1000+ hours of video lectures, mock tests, live doubt solving and more. You can signup or download Toppr app for Android or iOS .

Download NCERT solutions for all subjects of Class 9 here .

NCERT Solutions for Class 9 Maths Chapterwise

Chapter 1 – Number Systems

Chapter 2 – Polynomials

Chapter 3 – Coordinate Geometry

Chapter 4 – Linear Equations in Two Variables

Chapter 5 – Introduction to Euclid’s Geometry

Chapter 6 – Lines and Angles

Chapter 7 – Triangles

Chapter 8 – Quadrilaterals

Chapter 9 – Area 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 NCERT Solutions Chapterwise

Ncert solutions for class 9 maths chapter 1 number systems.

Rational numbers, Irrational numbers, Irrational numbers between two number are the topics covered under this chapter. We have solved more than 20 problems in this chapter. Download NCERT solutions for this chapter here .

NCERT Solutions for Class 9 Maths Chapter 2 Polynomials

In this chapter, you will learn more about polynomials, degree of polynomials, and theorem related to it. We have solved more than 70 problems in this chapter. Download NCERT Solutions for this chapter here .

NCERT Solutions for Class 9 Maths Chapter 3 Coordinate Geometry

In this chapter, we have covered Distance formula, section formula, and more about the area of triangles and quadrilaterals. Download NCERT solutions for this chapter here .

NCERT Solutions for Class 9 Maths Chapter 4 Linear Equations in Two Variables

We covered Linear equations in one variable in the previous Class, now it is time to solve two variables in equations. We have solved more than 30 problems in this chapter. Download NCERT Solutions for this chapter here .

NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry

Euclid’s postulates, Axioms, Conjectures are the topics covered under this chapter. Download NCERT solutions for this chapter here .

NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles

In this chapter, you will learn about Geometry, Lines, and different types of angles. Geometry is one of the most important parts of Mathematics and NCERT solutions will help you understand it easily. Download NCERT Solutions for this chapter here .

NCERT Solutions for Class 9 Maths Chapter 7 Triangles

In this chapter, you will learn about the topics like properties of triangles, Pythagoras theorem, Basic proportionality theorem and more. This is one of the important chapters in CBSE Class 9 Maths. We have solved more than 50 solutions for this chapter. Download CBSE NCERT solutions for this chapter here .

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

In this chapter, you will learn about Quadrilaterals and its properties and some theorems related to the quadrilateral. We have solved more than 50 solutions for this chapter. Download CBSE NCERT solutions for this chapter here .

NCERT Solutions for Class 9 Maths Chapter 9 Area of Parallelograms and Triangles

In this chapter, we will learn about calculating the area of triangles and parallelograms. Download NCERT solutions for this chapter here .

NCERT Solutions for Class 9 Maths Chapter 10 Circles

In this chapter, you will learn about the Arc of circle, tangents, cyclic quadrilateral and more. Download NCERT solutions for this chapter here .

NCERT Solutions for Class 9 Maths Chapter 11 Constructions

In this chapter, you will learn about the constructions of triangles, segments, tangent to a circle and more. Download NCERT solutions for this chapter here .

NCERT Solutions for Class 9 Maths Chapter 12 Heron’s Formula

Heron’s formula will help in measuring the surface area of Triangles. We have solved more than 20 problems in this chapter. Download NCERT Solutions for this chapter here .

NCERT Solutions for Class 9 Maths Chapter 13 Surface Areas and Volumes

In this chapter, you will learn about the surface area and volumes of different shapes like Cuboid, cubes, cylinder, etc. This chapter will help you to understand how to measure the dimensions of different shapes. Download CBSE NCERT solutions for this chapter here .

NCERT Solutions for Class 9 Maths Chapter 14 Statistics

Graphs, Charts, Frequency Distribution, Mean-Median-Mode are the topics covered under this chapter. This chapter will also help you in understand some of the business mathematics concepts. Download NCERT Solutions for this chapter here .

NCERT Solutions for Class 9 Maths Chapter 15 Probability

Probability is a complicated concept and also one of the most important. Probability is a topic which will also help you in solving real-life problems, business mathematics, and also important for many competitive entrance exams. Download NCERT Solutions for this chapter here .

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Solved Questions For You:

Question 1. State whether the following statements are true or false. Give reasons for your answers.

(i) Every natural number is a whole number. (ii) Every integer is a whole number. (iii) Every rational number is a whole number.

(i) True every natural is a whole number because natural number start from     1  and whole number start from  0 , so every natural number is a whole number.

Question 2. Write three numbers whose decimal expansions are non-terminating non-recurring.

Answer:  All irrational numbers have non-terminating and non-recurring decimal expression.

So, \(\sqrt{2}\), \(\sqrt{3}\), and \(\sqrt{5}\) will give such expressions.

Question 3.  Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.

Answer:  No,

for example \(\sqrt{4}\) = 2 is a rational number.

Question 4. Is zero a rational number? Can you write it in the form \(\frac{p}{q}\) ​ , where  p  and  q  are integers and  q \(\neq\)  = 0 ?

We know that a rational number is a number which can be written in the form of \(\frac{p}{q}\) , where  q  is not equal to zero. 

Thus, yes,  zero is a rational number, as it can be written in the form of \(\frac{p}{q}\) .

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NCERT Solutions for Class 9

• NCERT Solutions for Class 9 History Chapter 3 Free PDF Download
• NCERT Solutions for Class 9 History Chapter 6 Free PDF Download
• NCERT Solutions for Class 9 Maths Chapter 15 Free PDF Download
• NCERT Solutions for Class 9 Maths Chapter 7 Free PDF Download
• NCERT Solutions for Class 9 Maths Chapter 14 Free PDF Download
• NCERT Solutions for Class 9 Maths Chapter 12 Free PDF Download
• NCERT Solutions for Class 9 Maths Chapter 1 Free PDF Download
• NCERT Solutions for Class 9 Maths Chapter 9 Free PDF Download
• NCERT Solutions for Class 9 Maths Chapter 4 Free PDF Download
• NCERT Solutions for Class 9 Maths Chapter 11 Free PDF Download

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NCERT Solutions for Class 9

Class 9 NCERT solutions refer to the solutions provided for the various subjects’ textbooks for class 9 published by the National Council of Educational Research and Training (NCERT) in Bharat. NCERT is an autonomous organization that develops and publishes textbooks for the Indian school curriculum. Class 9 is an important year in a student’s education in Bharat, and Tiwari Academy NCERT solutions are designed to assist students in understanding and solving the exercises and problems presented in their class 9 textbooks. These solutions offer step-by-step explanations and answers for each exercise, chapter, and topic, helping students grasp concepts and improve their problem-solving skills. NCERT solutions for class 9 typically cover subjects like:

Mathematics Science Social Science English Hindi Sanskrit

For example, Class 9 Mathematics Tiwari Academy NCERT solutions cover topics such as Number Systems, Polynomials, Linear Equations in Two Variables, Coordinate Geometry, Introduction to Euclid’s Geometry, Lines and Angles, Triangles, Quadrilaterals, Circles, Surface Areas and Volumes, and Statistics and Probability.

Class 9 all Subjects Apps These solutions are widely used by students and teachers in Bharat to prepare for class 9 exams, as they are aligned with the curriculum prescribed by various education boards, including the Central Board of Secondary Education (CBSE). NCERT solutions help students understand key concepts, practice problem-solving, and ensure a strong foundation in their subjects. They are available in various formats, including printed books, online PDFs, and digital resources, making them accessible to a wide range of learners.

The syllabus of Class 9 Maths, Science and Social Science is so quite difficult. Students get confused to understand. This subject syllabus is also useful for competitive exam like IIT JEE, Olympiads, NEET etc. So you need to prepare and perform well in their CBSE 9 exams. These Subject wise NCERT Solutions for Class 9 are beneficial for students by practising with it they can get more marks in the final exam.

A team of respective subject teachers prepares the NCERT Solutions for Class 9. The step by step solutions is well explained in such a way that class 9 students can easily understand. In UP Board, the students are using NCERT Textbooks for Class 9 Maths, Science, Social Science and English. So, here they can download UP Board Solutions for the subjects Science, Maths, sst and English of 9th standard in Hindi Medium and English Medium.

Subject wise NCERT Solutions for Class 9 PDFs are available here with FREE of cost. NCERT Class 9 Solutions helps students in their final exam preparation as well. These solutions explain each chapter of all subjects in a simple and easily understandable way.

You can find solutions for all the subjects NCERT Class 9 Maths, Science, Hindi, English and Social Science books. All solutions are as per standard textbooks which are prescribed for CBSE students. Students who are worried about finding the solutions for all subjects click the link and learn both offline and online for all class 9 subjects.

NCERT Solutions for Class 9 Maths

Chapter Wise NCERT Solutions for Class 9 Maths FREE PDFs are available here for both medium English and Hindi. Here you can get all NCERT Maths textbook for Class 9th question answers for all chapters. Well, skilled Maths teacher review all chapter, including all exercises so that students can get accurate solutions. Chapter 1. Number Systems Chapter 2. Polynomials Chapter 3. Coordinate Geometry Chapter 4. Linear Equations in Two Variables Chapter 5. Introduction to Euclid’s Geometry Chapter 6. Lines and Angles Chapter 7. Triangles Chapter 8. Quadrilaterals Chapter 9. Circles Chapter 10. Heron’s Formula Chapter 11. Surface Areas and Volumes Chapter 12. Statistics So download NCERT Solutions for Class 9 Maths chapter wise. Students can practice all types of questions of the Maths chapters. Students need to do practice well to score good marks in the final exam, and also it helps to build a foundation for higher-level classes. You can get NCERT Solutions for Class 9 Maths for all chapters such as Number System, Polynomials, Introduction to Euclid’s Geometry, Triangles, Constructions, Surface Areas and Volumes, Probability and many more.

NCERT Solutions for Class 9 Science

Download Chapter wise PDF for NCERT Solutions Class 9 Science for both Hindi and English medium. All NCERT Solutions for Class 9 Science are here FREE of cost. Class 9 Science is also an important and difficult subject; students get confused with textbook questions. To find the solutions here and start practices regularly. These well-detailed answers clear all chapter related doubts and confusion. Chapter 1. Matter in Our Surroundings Chapter 2. Is Matter Around Us Pure Chapter 3. Atoms and Molecules Chapter 4. Structure of the Atom Chapter 5. The Fundamental Unit of Life Chapter 6. Tissues Chapter 7. Motion Chapter 8. Force and Laws of Motion Chapter 9. Gravitation Chapter 10. Work and Energy Chapter 11. Sound Chapter 12. Improvement in Food Resources These CBSE Class 9 Science NCERT Solutions cover Text Book solutions to all the textbook chapters included: Matter in Our Surroundings, Matter around Us Pure, Atoms and Molecules, Tissues, Force and Laws of Motion, Work and Energy, Improvement in Food Resources and many more. Other than these solutions here you can get other questions, exemplar problems, Worksheets, MCQ’s, Short and long answering questions etc.

NCERT Solutions for Class 9 Hindi

Get Chapter Wise NCERT Solutions for Class 9 Hindi  free of cost in both online and PDF format. Experienced Hindi Teacher provides all the chapters of NCERT Hindi Class 9 Textbook for the students.

NCERT Solutions for Class 9 Sanskrit

Students who are in class 9 and having CBSE syllabus can get the benefits of these Chapter Wise NCERT Solutions to get more marks in Sanskrit. Chapter 1. Bhaaratavaasantageeti Chapter 2. Swarnkakah Chapter 3. Godohanam Chapter 4. Sooktimuktikaam Chapter 5. Bhranto Balah Chapter 6. Lauhtula Chapter 7. Siktasetu Chapter 8. Jataayo Shauryam Chapter 9. Paryaavaranam Chapter 10. Vaangmanah Praanasvarupam The chapters covered in these solutions are based on the latest syllabus.

NCERT Solutions for Class 9 English

Students who are searching for the Chapter Wise NCERT Solutions for Class 9 English, then you are in the right place. Here you can get solutions for all chapters in subjects English with an easy language. You can download Chapter Wise NCERT Solutions for Class 9 English PDFs with FREE of cost. All the solutions are well written by the expert English teacher. Students will have a good practice of reading and writing with this language solution. Chapter Wise Solutions of Class 9 English Beehive Chapter 1. The Fun They Had Chapter 2. The Sound of Music Chapter 3. The Little Girl Chapter 4. A Truly Beautiful Mind Chapter 5. The Snake and the Mirror Chapter 6. My Childhood Chapter 7. Reach for the Top Chapter 8. Kathmandu Chapter 9. If I were you Chapter Wise Solutions of Class 9 English Moments Chapter 1. The Lost Child Chapter 2. The Adventures of Toto Chapter 3. Iswaran the Storyteller Chapter 4. In the Kingdom of Fools Chapter 5. The Happy Prince Chapter 6. Weathering the Storm in Ersama Chapter 7. The Last Leaf Chapter 8. A House is not a Home Chapter 9. The Beggar Chapters covered in this solution are The Fun They Had & The Road Not Taken, The Sound of Music & Wind, My Childhood & No Men Are Foreign, Reach for the Top & On Killing a Tree, Kathmandu & A Slumber Did My Spirit Seal If I were and many more. Just click the below-mentioned link as per your requirements.

NCERT Solutions for Class 9 Social Science

These FREE Chapter Wise NCERT Solutions for Class 9 Social Science are curated by the expert Social Science teacher. Here you can find step by step solutions to each questions chapter wise. Well answered solutions help you in fetching more marks. Students who feel shy and hesitate to ask answers to the class teacher can solve their problem with these solutions. Students can improve their performance by practice these solutions and get maximum marks in the final exam too. It covers all chapters such as The French Revolution, Socialism in Europe and the Russian Revolution, Forest Society and Colonialism, Pastoralists in the Modern World, India–Size and Location, Geography Chapter 3 Drainage, Natural Vegetation and Wildlife, The Story of Village Palampur etc.

About Class 9 Mathematics

In class 9 Geometry starts from chapter 5 ‘Introduction to Euclid’s Geometry, and continues upto chapter 11 ‘Constructions’. About 50% part of the Class 9 Maths is covered by Geometry only. Therefore, to get command over Mathematics in Class IX, the geometrical part should be done properly. If each questions and examples are done by self-practice (not by rote learning), the student will be confident in entire Maths in Class 9.

The way to study Science in Class 9

The expansion of Science subjects is beyond our assumption. So, no book is perfect to provide the complete information about any fact. That is why, initially the science is divided into three basic segments like Physics, Chemistry and Biology. Later on these segments further divided into so many parts to study in detail a particular branch of Physics, Chemistry or Bio. In class 9, NCERT is perfect to get all knowledge at this standard. All the chapters are properly described and the in text questions are given for practice. If students need more practice, they should refer NCERT Exemplar Problems book. There is no need so many extra books to increase unnecessary burden.

There are four sections in Class 9 Social Science as History, Geography, Political Science (Civics) and Economics. Books are as follows: Textbook in History for Class IX – ‘India and the Contemporary World – I’ Textbook in Geography for Class IX – ‘Contemporary India – I’ Textbook in Economics for Class IX – ‘Economics’ Textbook in Political Science for Class IX – ‘Democratic Politics – I’

How can I download Class 9 Solutions in PDF?

Class 9 Solutions are available on many website. Visit to Tiwari Academy to download solutions in PDF. On other websites you need to register your mobile number or proper registration is required but on Tiwari Academy, there is no need of registration.

Is Maths solutions for class 9 available in Hindi and English Medium?

Yes, Maths for class 9 is available in both English and Hindi Medium.

Why should I use Tiwari Academy Class 9 NCERT Solutions?

Tiwari Academy is maintaining a user friendly and trouble free website for the students. There is no login, no registration of phone number, no business messaging or no call for selling products on Tiwari Academy.

Are NCERT Solutions for Class 9 all subjects available on Tiwari Academy?

Tiwari Academy is providing class 9 all subjects NCERT Solutions in English and Hindi Medium.

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• Class 9 Math NCERT Exemplar Chapter wise Solutions - Free PDF Download
• Textbook Solutions

NCERT Exemplar Solutions for Class 9

NCERT Exemplar book for Class 9 is a questionnaire book curated by NCERT. This book is made in such a way so that students can use their high order thinking skills while solving examples. In this book, they will find some interesting different types of questions under the syllabus. Vedantu helps you to understand and learn all the answers to the provided questions of the NCERT Exemplar book. Vedantu guides you through the solutions given in the study materials and questionnaires prepared by an excellent team of expert teachers. Hence, keep it in mind to download NCERT Solutions PDF and start learning. You can download NCERT Solutions for Class 9 Maths and NCERT Solution Class 9 Science to help you to revise the complete Syllabus and score more marks in your examinations.

NCERT Exemplar for Class 9 Maths

Chapters in NCERT Exemplar for Class 9- Math

The chapters are as follows:

FAQs on Class 9 Math NCERT Exemplar Chapter wise Solutions - Free PDF Download

1. Are the Solutions Present in Vedantu Perfect According to the Exam Point of View?

Yes, the solutions present in Vedantu are according to the exam point of view. The solution is compiled and written by the teachers of the respective subjects to help every student understand it easily. Every approach towards every solution is made to strengthen the student’s knowledge about the topic.

Vedantu gives you the best service when it comes to providing solutions.

2. Are Vedantu’s Resources Beneficial for Exam Preparation?

Yes, Vedantu’s resources are beneficial for exam preparation. Every teacher has left no stone unturned for preparing the finest study materials and solutions as the students’ need. Without covering the unnecessary topics, they have made a perfect study material just covering what you need to score better in future exams. A perfect education site to trust. Vedantu is someone who will never break your trust. It’s up to the mark.

3. How do I practice Math for Class 9 NCERT?

You can solve a lot of practice papers so as to get an overall view of what might come for the examinations. Regular practice will help you in solving even the most challenging of all problems. NCERT Exemplar book for Class 9 Math has some really interesting questions for you to work upon. They have been framed in accordance with the NCERT guidelines and are useful for practice. These books can be found on Vedantu’s portal and you can refer to them whenever you wish to. 

4. Are Polynomials important for Class 9  NCERT Math?

All chapters of NCERT Class 9 Math including the one on Polynomials is extremely important. Polynomials relate to the expression between coefficient and variables and have more than two terms irrespective of the operators.

NCERT Exemplar Class 9 Mathematics has sums that will enable students to grasp the concept of polynomials in the best manner possible. Polynomials are quite significant as a chapter as they will reappear in future Math studies as well.  Students must make sure that no chapters are skipped.

5. Is Vedantu a reliable site for online studies?

Yes, Vedantu is indeed a reliable site when it comes to online studies. The study materials present on its portal have been designed in consultation with subject matter experts and so, only the most relevant portions have been included on it. Anything that is unnecessary or outside of the syllabus is not present on Vedantu. Trust is an important factor in academics as the very basis of children’s future stands on it. Vedantu can be completely trusted in this regard as it will never let you down. 

6. Where can I find the chapter on Triangles for NCERT Class 9 Math? 

Students can find this particular chapter in the NCERT Exemplar book for Class 9 which is a questionnaire book made by the NCERT. In this book, they will be able to find all the pertinent topics for Class 9 which they might have to prepare for. The unique sort of questions solved in the book and the explanations given are sufficient to take assessments. You can find this book on Vedantu . A summary of each of the Math chapters has been provided along with the important formulae.

7. What is Probability?

It is the measure of the likelihood of any event that may occur and can range between 0 and 1. 0 probability indicates that there are no chances of it occurring while the probability of 1 indicates that there is some chance that the event might take place.

The explanations along with valid examples for each have been provided in the NCERT Exemplar book for Class 9 Math. The set of questions asked covers the broad topic at hand and provides students with overall learning of the most crucial topics. One must practice all the solutions that are present in the book. 

CBSE CLASS 9 STUDY MATERIALS

CBSE NCERT Solutions

NCERT and CBSE Solutions for free

Class 9 Mathematics Circle Assignments

We have provided below free printable Class 9 Mathematics Circle Assignments for Download in PDF. The Assignments have been designed based on the latest NCERT Book for Class 9 Mathematics Circle . These Assignments for Grade 9 Mathematics Circle cover all important topics which can come in your standard 9 tests and examinations. Free printable Assignments for CBSE Class 9 Mathematics Circle , school and class assignments, and practice test papers have been designed by our highly experienced class 9 faculty. You can free download CBSE NCERT printable Assignments for Mathematics Circle Class 9 with solutions and answers. All Assignments and test sheets have been prepared by expert teachers as per the latest Syllabus in Mathematics Circle Class 9. Students can click on the links below and download all Pdf Assignments for Mathematics Circle class 9 for free. All latest Kendriya Vidyalaya Class 9 Mathematics Circle Assignments with Answers and test papers are given below.

Mathematics Circle Class 9 Assignments Pdf Download

We have provided below the biggest collection of free CBSE NCERT KVS Assignments for Class 9 Mathematics Circle . Students and teachers can download and save all free Mathematics Circle assignments in Pdf for grade 9th. Our expert faculty have covered Class 9 important questions and answers for Mathematics Circle as per the latest syllabus for the current academic year. All test papers and question banks for Class 9 Mathematics Circle and CBSE Assignments for Mathematics Circle Class 9 will be really helpful for standard 9th students to prepare for the class tests and school examinations. Class 9th students can easily free download in Pdf all printable practice worksheets given below.

Topicwise Assignments for Class 9 Mathematics Circle Download in Pdf

Advantages of Class 9 Mathematics Circle Assignments

• As we have the best and largest collection of Mathematics Circle assignments for Grade 9, you will be able to easily get full list of solved important questions which can come in your examinations.
• Students will be able to go through all important and critical topics given in your CBSE Mathematics Circle textbooks for Class 9 .
• All Mathematics Circle assignments for Class 9 have been designed with answers. Students should solve them yourself and then compare with the solutions provided by us.
• Class 9 Students studying in per CBSE, NCERT and KVS schools will be able to free download all Mathematics Circle chapter wise worksheets and assignments for free in Pdf
• Class 9 Mathematics Circle question bank will help to improve subject understanding which will help to get better rank in exams

Frequently Asked Questions by Class 9 Mathematics Circle students

At https://www.cbsencertsolutions.com, we have provided the biggest database of free assignments for Mathematics Circle Class 9 which you can download in Pdf

We provide here Standard 9 Mathematics Circle chapter-wise assignments which can be easily downloaded in Pdf format for free.

You can click on the links above and get assignments for Mathematics Circle in Grade 9, all topic-wise question banks with solutions have been provided here. You can click on the links to download in Pdf.

We have provided here topic-wise Mathematics Circle Grade 9 question banks, revision notes and questions for all difficult topics, and other study material.

We have provided the best collection of question bank and practice tests for Class 9 for all subjects. You can download them all and use them offline without the internet.

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NCERT Book for Class 9th Maths PDF (2024-25)

Ncert book for class 9 maths: download the latest class 9th maths ncert book in a chapter-wise pdf for the current academic session 2024-25. .

NCERT Book for Class 9 Maths: Get the latest NCERT Book for Class 9 Maths for the 2024-25 session for English and Hindi medium. The NCERT Book for Class 9 Maths is revised by the National Council of Educational Research and Training (NCERT) where they have rationalised the contents of the old textbook. Students must check here the names of chapters and topics that have been deleted from the old book and are not relevant for the current academic session. Download the latest and revised book to study Mathematics effectively and perform well in your class tests and the annual CBSE Class 9 Maths Exam 2024-25. 

NCERT Book for Class 9 Maths (2024-25) - in English

List of rationalised content in class 9 maths ncert book.

CBSE Class 9 Maths Deleted Syllabus

NCERT Book for Class 9 Maths  (2023-24) -  in Hindi

Check detailed ncert solutions for class 9 maths (by subject experts).

Why you should read NCERT Book for Class 9 Maths?

NCERT books are best known for presenting the concepts most simply so that students can grasp them effortlessly. Therefore, students are always recommended to use NCERT books to excel in their studies.

• Designed to offer in-depth knowledge in easy and simple language.
• Strictly adheres to the CBSE curriculum which is considered to be sufficient enough to form the base for the annual examinations. 
• Ideal for a comprehensive study to develop a clear concept. 
• Includes several questions of different formats which are perfect to test students' understanding of concepts.

You can also get the latest NCERT Books and exclusive NCERT Solutions for all other subjects of Class 9 by clicking on the following links:

NCERT Books for Class 9  - All Subjects

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• Where can I get PDF of Class 9 Maths NCERT Book? + Get free PDF download for the latest and revised edition of Class 9 Maths NCERT Book from this article of Jagran Josh. You will get NCERT Book in English and Hindi languages here.
• Is Maths easy in class 9? + Class 9 Mathematics has some complex topics which can be challenging for some students. However, regular practice and perseverance can contribute to making mathematics more manageable for all. A positive approach towards the subjects helps make it less daunting.
• Which book is best for class 9 Maths exam? + NCERT book is considered the best book for clearing all concepts of Class 9 Maths. The book offers several advantages, including a comprehensive coverage of the syllabus, clear explanations, and a variety of practice problems to enhance understanding and problem-solving skills.
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• NCERT Solutions
• NCERT Class 9
• NCERT 9 Maths
• Chapter 6: Lines And Angles

NCERT Solutions for Class 9 Maths Chapter 6 - Lines And Angles

Ncert solutions class 9 maths chapter 6 – cbse free pdf download.

NCERT Solutions for Class 9 Maths Chapter 6, Lines and Angles, deals with the questions and answers related to the chapter Lines and Angles. This topic introduces you to basic Geometry, primarily focusing on the properties of the angles formed: i) when two lines intersect each other and ii) when a line intersects two or more parallel lines at distinct points. This chapter falls under the Unit – Geometry and is included in the CBSE Syllabus of Class 9 Maths. Students can ace well in the final exams with the help of NCERT Solutions given at BYJU’S.

Download Exclusively Curated Chapter Notes for Class 9 Maths Chapter – 6 Lines And Angles

Download most important questions for class 9 maths chapter – 6 lines and angles.

Now, you must be wondering why we are studying Lines and Angles. What are its real-life applications of it? The answer is that lines and angles are everywhere around us. The architecture uses lines and angles to design the structure of a building. When you stop at a signal and then move on when the signal light is green, then you either take a left-angle turn or right-angle turn or move in a straight line. When you have to find the height of a tower or the location of an aircraft, you need to know the angles. In NCERT Solutions for Class 9 Maths Chapter 6, you will learn to solve the questions related to all the concepts of Lines and Angles as per the updated CBSE syllabus for 2023-24. Students can avail of a complete and free PDF of this chapter’s NCERT Solutions for Class 9 Maths  from the link given below.

• 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 6 – Lines and Angles

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Access Answers of Maths NCERT Class 9 Maths Chapter 6 – Lines And Angles

List of Exercises in Class 9 Maths Chapter 6

Exercise 6.1 Solutions 6 Questions (5 Short Answer Questions, 1 Long Answer Question) Exercise 6.2 Solutions 6 Questions (3 Short Answer Questions, 3 Long Answer Questions) Exercise 6.3 Solutions 6 Questions (5 Short Answer Questions, 1 Long Answer Question)

Class 9 Maths Chapter 6 Exercise: 6.1 (Page No: 96)

Exercise: 6.1 (Page No: 96)

1. In Fig. 6.13, lines AB and CD intersect at O. If ∠AOC +∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex ∠COE.

From the diagram, we have

(∠AOC +∠BOE +∠COE) and (∠COE +∠BOD +∠BOE) forms a straight line.

So, ∠AOC+∠BOE +∠COE = ∠COE +∠BOD+∠BOE = 180°

Now, by putting the values of ∠AOC + ∠BOE = 70° and ∠BOD = 40° we get

∠COE = 110° and ∠BOE = 30°

So, reflex ∠COE = 360 o – 110 o = 250 o

2. In Fig. 6.14, lines XY and MN intersect at O. If ∠POY = 90° and a : b = 2 : 3, find c.

We know that the sum of linear pair is always equal to 180°

∠POY +a +b = 180°

Putting the value of ∠POY = 90° (as given in the question), we get,

Now, it is given that a:b = 2:3, so

Let a be 2x and b be 3x

∴ 2x+3x = 90°

Solving this, we get

So, x = 18°

∴ a = 2×18° = 36°

Similarly, b can be calculated, and the value will be

b = 3×18° = 54°

From the diagram, b+c also forms a straight angle, so

c+54° = 180°

3. In Fig. 6.15, ∠PQR = ∠PRQ, then prove that ∠PQS = ∠PRT.

Since ST is a straight line, so

∠ PQS+ ∠ PQR = 180° (linear pair) and

∠ PRT+ ∠ PRQ = 180° (linear pair)

Now,  ∠ PQS +  ∠ PQR =  ∠ PRT+ ∠ PRQ = 180°

Since  ∠ PQR = ∠ PRQ (as given in the question)

∠ PQS =  ∠ PRT. (Hence proved).

4. In Fig. 6.16, if x+y = w+z, then prove that AOB is a line.

To prove AOB is a straight line, we will have to prove x+y is a linear pair

i.e. x+y = 180°

We know that the angles around a point are 360°, so

x+y+w+z = 360°

In the question, it is given that,

So, (x+y)+(x+y) = 360°

2(x+y) = 360°

∴ (x+y) = 180° (Hence proved).

5. In Fig. 6.17, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that ∠ROS = ½ (∠QOS – ∠POS).

In the question, it is given that (OR ⊥ PQ) and ∠POQ = 180°

We can write it as ∠ROP = ∠ROQ = 90 0

We know that

∠ROP = ∠ROQ

It can be written as

∠POS + ∠ROS = ∠ROQ

∠POS + ∠ROS = ∠QOS – ∠ROS

∠SOR + ∠ROS = ∠QOS – ∠POS

2∠ROS = ∠QOS – ∠POS

Or, ∠ROS = 1/2 (∠QOS – ∠POS)(Hence proved).

6. It is given that ∠XYZ = 64° and XY is produced to point P. Draw a figure from the given information. If ray YQ bisects ∠ZYP, find ∠XYQ and reflex ∠QYP.

Here, XP is a straight line

So, ∠XYZ +∠ZYP = 180°

Putting the value of ∠XYZ = 64°, we get

64° +∠ZYP = 180°

∴ ∠ZYP = 116°

From the diagram, we also know that ∠ZYP = ∠ZYQ + ∠QYP

Now, as YQ bisects ∠ZYP,

∠ZYQ = ∠QYP

Or, ∠ZYP = 2∠ZYQ

∴ ∠ZYQ = ∠QYP = 58°

Again, ∠XYQ = ∠XYZ + ∠ZYQ

By putting the value of ∠XYZ = 64° and ∠ZYQ = 58°, we get.

∠XYQ = 64°+58°

Or, ∠XYQ = 122°

Now, reflex ∠QYP = 180°+XYQ

We computed that the value of ∠XYQ = 122°.

∠QYP = 180°+122°

∴ ∠QYP = 302°

Exercise: 6.2 (Page No: 103)

1. In Fig. 6.28, find the values of x and y and then show that AB || CD.

We know that a linear pair is equal to 180°.

So, x+50° = 180°

We also know that vertically opposite angles are equal.

So, y = 130°

In two parallel lines, the alternate interior angles are equal. In this,

x = y = 130°

This proves that alternate interior angles are equal, so AB || CD.

2. In Fig. 6.29, if AB || CD, CD || EF and y : z = 3 : 7, find x.

It is known that AB || CD and CD||EF

As the angles on the same side of a transversal line sum up to 180°,

x + y = 180° —–(i)

∠O = z (Since they are corresponding angles)

and, y +∠O = 180° (Since they are a linear pair)

So, y+z = 180°

Now, let y = 3w and hence, z = 7w (As y : z = 3 : 7)

∴ 3w+7w = 180°

Or, 10 w = 180°

So, w = 18°

Now, y = 3×18° = 54°

and, z = 7×18° = 126°

Now, angle x can be calculated from equation (i)

Or, x+54° = 180°

3. In Fig. 6.30, if AB || CD, EF ⊥ CD and ∠GED = 126°, find ∠AGE, ∠GEF and ∠FGE.

Since AB || CD, GE is a transversal.

It is given that ∠GED = 126°

So, ∠GED = ∠AGE = 126° (As they are alternate interior angles)

∠GED = ∠GEF +∠FED

As EF⊥ CD, ∠FED = 90°

∴ ∠GED = ∠GEF+90°

Or, ∠GEF = 126° – 90° = 36°

Again, ∠FGE +∠GED = 180° (Transversal)

Putting the value of ∠GED = 126°, we get

∠AGE = 126°

∠GEF = 36° and

4. In Fig. 6.31, if PQ || ST, ∠PQR = 110° and ∠RST = 130°, find ∠QRS.

[Hint : Draw a line parallel to ST through point R.]

First, construct a line XY parallel to PQ.

We know that the angles on the same side of the transversal is equal to 180°.

So, ∠PQR+∠QRX = 180°

Or, ∠QRX = 180°-110°

∴ ∠QRX = 70°

∠RST +∠SRY = 180°

Or, ∠SRY = 180°- 130°

∴ ∠SRY = 50°

Now, for the linear pairs on the line XY-

∠QRX+∠QRS+∠SRY = 180°

Putting their respective values, we get

∠QRS = 180° – 70° – 50°

Hence, ∠QRS = 60°

5. In Fig. 6.32, if AB || CD, ∠APQ = 50° and ∠PRD = 127°, find x and y.

From the diagram,

∠APQ = ∠PQR (Alternate interior angles)

Now, putting the value of ∠APQ = 50° and ∠PQR = x, we get

∠APR = ∠PRD (Alternate interior angles)

Or, ∠APR = 127° (As it is given that ∠PRD = 127°)

∠APR = ∠APQ+∠QPR

Now, putting values of ∠QPR = y and ∠APR = 127°, we get

127° = 50°+ y

Or, y = 77°

Thus, the values of x and y are calculated as:

x = 50° and y = 77°

6. In Fig. 6.33, PQ and RS are two mirrors placed parallel to each other. An incident ray AB strikes the mirror PQ at B, the reflected ray moves along the path BC and strikes the mirror RS at C and again reflects back along CD. Prove that AB || CD.

First, draw two lines, BE and CF, such that BE ⊥ PQ and CF ⊥ RS.

Now, since PQ || RS,

So, BE || CF

We know that,

Angle of incidence = Angle of reflection (By the law of reflection)

∠1 = ∠2 and

We also know that alternate interior angles are equal. Here, BE ⊥ CF and the transversal line BC cuts them at B and C

So, ∠2 = ∠3 (As they are alternate interior angles)

Now, ∠1 +∠2 = ∠3 +∠4

Or, ∠ABC = ∠DCB

So, AB || CD (alternate interior angles are equal)

Exercise: 6.3 (Page No: 107)

1. In Fig. 6.39, sides QP and RQ of ΔPQR are produced to points S and T,respectively. If ∠SPR = 135° and ∠PQT = 110°, find ∠PRQ.

It is given the TQR is a straight line, and so, the linear pairs (i.e. ∠TQP and ∠PQR) will add up to 180°

So, ∠TQP +∠PQR = 180°

Now, putting the value of ∠TQP = 110°, we get

Consider the ΔPQR,

Here, the side QP is extended to S, and so ∠SPR forms the exterior angle.

Thus, ∠SPR (∠SPR = 135°) is equal to the sum of interior opposite angles. (Triangle property)

Or, ∠PQR +∠PRQ = 135°

Now, putting the value of ∠PQR = 70°, we get

∠PRQ = 135°-70°

Hence, ∠PRQ = 65°

2. In Fig. 6.40, ∠X = 62°, ∠XYZ = 54°. If YO and ZO are the bisectors of ∠XYZ and ∠XZY, respectively of Δ XYZ, find ∠OZY and ∠YOZ.

We know that the sum of the interior angles of the triangle.

So, ∠X +∠XYZ +∠XZY = 180°

Putting the values as given in the question, we get

62°+54° +∠XZY = 180°

Or, ∠XZY = 64°

Now, we know that ZO is the bisector, so

∠OZY = ½ ∠XZY

∴ ∠OZY = 32°

Similarly, YO is a bisector, so

∠OYZ = ½ ∠XYZ

Or, ∠OYZ = 27° (As ∠XYZ = 54°)

Now, as the sum of the interior angles of the triangle,

∠OZY +∠OYZ +∠O = 180°

∠O = 180°-32°-27°

Hence, ∠O = 121°

3. In Fig. 6.41, if AB || DE, ∠BAC = 35° and ∠CDE = 53°, find ∠DCE.

We know that AE is a transversal since AB || DE

Here ∠BAC and ∠AED are alternate interior angles.

Hence, ∠BAC = ∠AED

It is given that ∠BAC = 35°

Now consider the triangle CDE. We know that the sum of the interior angles of a triangle is 180°.

∴ ∠DCE+∠CED+∠CDE = 180°

Putting the values, we get

∠DCE+35°+53° = 180°

Hence, ∠DCE = 92°

4. In Fig. 6.42, if lines PQ and RS intersect at point T, such that ∠PRT = 40°, ∠RPT = 95° and ∠TSQ = 75°, find ∠SQT.

Consider triangle PRT.

∠PRT +∠RPT + ∠PTR = 180°

So, ∠PTR = 45°

Now ∠PTR will be equal to ∠STQ as they are vertically opposite angles.

So, ∠PTR = ∠STQ = 45°

Again, in triangle STQ,

∠TSQ +∠PTR + ∠SQT = 180°

74° + 45° + ∠SQT = 180°

5. In Fig. 6.43, if PQ ⊥ PS, PQ || SR, ∠SQR = 28° and ∠QRT = 65°, then find the values of x and y.

x +∠SQR = ∠QRT (As they are alternate angles since QR is transversal)

So, x+28° = 65°

It is also known that alternate interior angles are the same, and so

∠QSR = x = 37°

∠QRS +∠QRT = 180° (As they are a Linear pair)

Or, ∠QRS+65° = 180°

So, ∠QRS = 115°

Using the angle sum property in Δ SPQ,

∠SPQ + x + y = 180°

90°+37° + y = 180°

y = 180 0 – 127 0 = 53 0

Hence, y = 53°

6. In Fig. 6.44, the side QR of ΔPQR is produced to a point S. If the bisectors of ∠PQR and ∠PRS meet at point T, then prove that ∠QTR = ½ ∠QPR.

Consider the ΔPQR. ∠PRS is the exterior angle, and ∠QPR and ∠PQR are the interior angles.

So, ∠PRS = ∠QPR+∠PQR (According to triangle property)

Or, ∠PRS -∠PQR = ∠QPR ———–(i)

Now, consider the ΔQRT,

∠TRS = ∠TQR+∠QTR

Or, ∠QTR = ∠TRS-∠TQR

We know that QT and RT bisect ∠PQR and ∠PRS, respectively.

So, ∠PRS = 2 ∠TRS and ∠PQR = 2∠TQR

Now, ∠QTR = ½ ∠PRS – ½∠PQR

Or, ∠QTR = ½ (∠PRS -∠PQR)

From (i), we know that ∠PRS -∠PQR = ∠QPR

So, ∠QTR = ½ ∠QPR (hence proved).

The Class 9 Maths theory paper is of 80 marks. Out of which, Geometry constitutes a total of 22 marks which includes Introduction to Euclid’s Geometry, Lines and Angles, Triangles, Quadrilaterals, Areas, Circles, and Constructions. You can see that it constitutes approximately 27% of weightage. So, using the NCERT Solutions for Class 9 , students can easily score high marks by having a thorough understanding of this topic.

Solve all the exercise problems of Lines and Angles. Refer to the NCERT Solutions of Class 9 provided by our Experts below. It will help you to solve the questions in an easy way.

Summary of NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles

Before starting to solve the exercise problems, you must first read the theory part and get to know the basic terms, definitions and theorems. After that, go through the solved examples of Lines and Angles that are given in the Class 9 NCERT Book . It will make your concepts more clear. Then you can start solving the exercise problems with the help of NCERT Solutions.

Key Points of NCERT Class 9 Maths Chapter 6 – Lines and Angles

After solving the Line and Angles chapter of Class 9 Maths, you will get to know the following points:

• Basic terms and definitions related to a line segment, ray, collinear points, non-collinear points, intersecting and non-intersecting lines.
• Pair of angles (reflex, complementary, supplementary, adjacent, vertical opposite, linear pair).
• Parallel and transversal lines and theorems related to them.
• Angle sum property of a triangle.

We hope this information on “ NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles” is useful for students. Stay tuned for further updates on CBSE and other competitive exams. To access interactive Maths and Science Videos, download BYJU’S App and subscribe to YouTube Channel.

To learn more concepts regarding Lines and Angles, students can access the other study materials which are prescribed by the CBSE board. By doing this, students will be able to solve the questions that would appear in the CBSE exams.

• RD Sharma Solutions for Class 9 Maths Chapter 8 Lines and Angles

Disclaimer:

Dropped Topics –  6.5 Parallel lines and a transversal and 6.7 Angle sum property of a triangle.

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