CBSE

# List of Most Important Theorems in Mathematics Class 10 for CBSE Board Exam 2024

One of the core and most important subjects in the school curriculum is maths. It supports students’ intellectual development. Additionally, mathematical theorems are taught to students, which increases brain activity and establishes the foundations of Mathematics.

The discipline of mathematics relies heavily on mathematical theorems, each branch of which has a wealth of established ideas proposed by internationally recognised mathematicians. To establish a solid foundation in fundamental mathematics, these theorems are essential.

A mathematical statement is only proved as a theorem if it has been demonstrated to be true. Proving these theorems reduces mathematical errors by confirming their validity and promoting the growth of logical reasoning and thinking abilities. The blog aims to review important theorems in mathematics class 10 from the NCERT syllabus. You need to be aware of how these theorems are proven to prepare for the exam.

## Overview of CBSE Class 10 Math Exam 2024

There are seven sections in this maths syllabus for class 10 CBSE 2023-24: algebra, coordinate geometry, geometry, trigonometry, mensuration, statistics, probability, and number systems. There are fifteen chapters in these units.

It is essential to thoroughly cover the entire CBSE Class 10 Maths syllabus 2023-24 if you want to do well on the exam.

There are seven units in the class 10 CBSE 2023–2024 maths curriculum. and there are additional topics and subtopics within these units. Before we delve into important theorems in mathematics class 10, let’s review the comprehensive math class 10 syllabus 2023–24, by units, topics, details and marking scheme.

20 marks are allocated for the internal assessments. They are structured as below:

## Importance of Theorem and its significance

A theorem’s importance comes from its ability to reveal fundamental facts and make crucial connections in the study of mathematics and other subjects. Strong instruments and theorems not only offer exacting proofs for particular claims but also advance our knowledge of underlying concepts and connections. There are several important theorems in mathematics class 10 CBSE. Let’s examine the value of the theorems in mathematics and how they benefit students.

1. Problem-Solving: By providing proof, they not only aid in problem-solving but also enable students to comprehend the underlying ideas more thoroughly.
2. Foundations of Mathematics: Once students have a firm understanding of the statements and their proofs, theorems serve as the cornerstone of fundamental mathematics. It helps them to develop deductive reasoning skills.
3. Understanding the Origins: Theorems are important concepts that are regarded as indestructible truths. In addition to making mathematical problems easier to solve, theorems and their proofs aid in the development of a deeper comprehension of the fundamental concepts.
4. Logical Reasoning: When students fully comprehend the statements and their proofs, theorems not only serve as the cornerstone of fundamental mathematics but also aid in the development of deductive reasoning.

## List of Important Theorems in Maths Class 10

The most significant theorems for Class 10 are as follows:

1. Circle Theorems
2. Triangle Theorems
3. Real Numbers
4. Euclid’s Division Lemma
5. Fundamental Theorem of Arithmetic
6. Polynomials: Remainder Theorem
7. Factor Theorem
8. Basic Proportionality Theorem
9. Baye’s Theorem

Let’s delve into each one of these in detail.

### 1. Circle Theorems

The circle is the subject of several theorems. It is one of the important theorems in mathematics class 10. Students in Classes 9 and 10 need to understand the circle theorems.

Theorem 1 states that equal chords subtend equal angles at the circle’s centre.

Theorem 1 Converse states that if two chords have equal angles subtended at their centres, then the chords have equal lengths.

Theorem 2 states that if a perpendicular is drawn from the circle’s centre, it will bisect the chord.

Theorem 2 Converse states that a line that cuts a chord in half through the centre of the circle is perpendicular to the chord.

Theorem 3 states that a circle’s equal chords are equally spaced (equidistant) from its centre.

Converse of Theorem 3 states that the length of a circle’s chords is equal when they are spaced equally from its centre.

Theorem 4 states that the measure of angles subtended from the same arc to any point on the circle’s circumference is equal to half of the angles subtended by the same arc at the centre.

Theorem 5 states that a cyclic quadrilateral’s opposing angles are supplementary.

Circle theorem is an integral part of the list of 10th-class important theorem. Students are advised to study them and prepare well.

### 2. Triangle Theorems

• If the corresponding angles and sides of two polygons with the same number of sides are equal, they are said to be similar.
• If the corresponding angles and sides of two triangles have the same ratio, they are said to be similar.
• Two triangles are similar if their corresponding sides have the same ratio and their corresponding angles are equal.
• When two triangles are similar to each other, their corresponding angles are equal because the sides of any one of the triangles are proportionate to the sides of the other.

Triangle theorems is one of the important theorems in mathematics class 10. The triangle theorem is an integral part of the list of 10th-class important theorem. Students are encouraged to thoroughly read and prepare for them. Thus, for students to ace the exams, it is important to be well-versed with triangle theorems.

### 3. Real Numbers

Let’s have a look at comprehensive details of Theorems in the Real Numbers.

Theorem 1:

If a and b are real numbers such that a+b=a, then b=0.

• Proof: Suppose that a+b=a. We will manipulate both sides of this equation to arrive at the conclusion that b=0.

(1)

a+b=a

(a+b)+(−a)=a+(−a)

b+(a+(−a))=a+(−a)

(b+a)+(−a)=a+(−a)

b+0=0

b=0

Theorem 2:

If a and b are real numbers such that a⋅b=a, then b=1.

• Proof: Suppose that a⋅b=a. We will manipulate both sides of this equation to arrive at the conclusion that b=1.

(2)

a⋅b=a

a(−1)⋅(a⋅b)=a(−1)⋅a

(a(−1)⋅a)⋅b=a(−1)⋅a

1⋅b=1b=1

Theorem 3:

If a is a real number then a⋅0=0.

• Proof: Let a∈R.

(3)

a⋅0=0

a+a⋅0=a+0

a⋅1+a⋅0=a+0

a⋅(1+0)=a+0

a⋅1=a+0

a=a

• Since we have shown that a+(a⋅0)=a+0 implies that a=a, then we must have that a⋅0=0.

Theorem 4:

If a and b are real numbers and a⋅b=0 then a=0 or b=0 or both a,b=0.

• Proof: If both a,b=0 then 0⋅0=0. Now without loss of generality, assume that a≠0.

(5)

a⋅b=0

a−1⋅(a⋅b)=a−1⋅0

a−1⋅(a⋅b)=0

(a−1⋅a)⋅b=0

1⋅b=0

b=0

• So if a≠0 then b=0.

Theorem 5:

If a and b are real numbers such that a+b=0 then b=−a.

• Proof: Let a,b∈R be such that a+b=0.

(6)

a+b=0

(−a)+(a+b)=(−a)+0

((−a)+a)+b=−a

0+b=−a

b=−a

Theorem 6:

If a is a real number, then −1⋅a=−a.

• Proof: Let a∈R. To show that −1⋅a=−a we will show that both values are additive inverses of a, and then by theorem 6 (which says that additive inverses are unique), we can conclude that −1⋅a=−a.
• First a+−1⋅a=1⋅a+−1⋅a=(1−1)⋅a=0⋅a=0. So a is the additive inverse of −1⋅a.
• Furthermore a+(−a)=0. Since additive inverses are unique we have that −1⋅a=−a.

Theorem 7:

If a is a real number then −(−a)=a.

• Proof: Let a∈R. Notice that (−a)−(−a)=[(−1)⋅a]−[(−1)⋅a]=(−1)⋅(a−a)=−1⋅0=0. So the additive inverse to −a is −(−a). But the additive inverse to −a is also a which implies that −(−a)=a.

Theorem 8:

(−1)⋅(−1)=1.

• Proof: Notice that (−1)⋅(−1)=−(−1)=1.

Theorem 9:

If a and b are real numbers then −(a+b)=(−a)+(−b).

• Proof: Let a,b∈R. Then −(a+b)=(−1)⋅(a+b)=(−1)⋅a+(−1)⋅b=(−a)+(−b).

Theorem 10:

If a and b are real numbers then (−a)⋅(−b)=a⋅b.

• Proof: Let a,b∈R. Then (−a)⋅(−b)=[(−1)⋅a]⋅[(−1)⋅b]=[(−1)⋅(−1)]⋅[a⋅b]=1⋅[a⋅b]=a⋅b.

### 4. Euclid’s Division Lemma

If we have two positive integers, a and b, then there exist unique integers, q and r, which satisfy the formula a = bq + r, where 0 ≤ r < b, according to Euclid’s Division Lemma.

Euclid’s Division Lemma is an integral part of the list of 10th class important theorems. Euclid’s division lemma serves as the foundation for the Euclidean division algorithm. Using Euclid’s division algorithm, we can find the Highest Common Factor (HCF) of two positive integers, a and b. The largest number that divides two or more positive integers exactly is the HCF. This indicates that there is no remainder when dividing both the integers a and b.

### 5. Fundamental Theorem of Arithmetic

The fundamental theorem of arithmetic states “factorization of every composite number can be expressed as a product of primes irrespective of the order in which the prime factors of that respective number occur”.

Let’s look up the prime factorization of 240, for instance.

We can calculate 240 = 2 × 2 × 2 × 2 × 3 × 5 using the above figure.

We also learn from this theorem that this factorization needs to be distinct. That is, 240 cannot be expressed as a product of primes in any other way. We can, of course, alter the sequence in which the prime factors appear. The prime factorization can be expressed, for instance, as follows: 240 = 31 × 24 × 51 or 31 × 22 × 51 × 22, etc. However, the set of prime factors is distinct, as is the frequency at which each factor appears.

In other words, there is only one prime factorization that could exist for 240, consisting of four factors of 2, or 24, one factor of 3, or 31, and one factor of 5, or 51.

### 6. Polynomials: Remainder Theorem

An approach to Euclidean division of polynomials is the Remainder Theorem. This theorem states that when a polynomial P(x) is divided by a factor (x – a), the result is a smaller polynomial and a remainder rather than a polynomial element. The remainder that was found is actually P(a), or the value of P(x) at x = a. In other words, x -a is P(x)’s divisor if and only if P(a) = 0. It is used to factor polynomials of all degrees in a sophisticated way.

For instance, when f(a) = a3-12a2-42 is divided by (a-3), a2-9a-27 is the quotient, and -123 is the remainder.

Putting a-3 = 0

so a = 3.

f(a) = f(3) = -123 as a result

As a result, the remainder theorem is satisfied.

### 7. Factor Theorem

A specific type of polynomial remainder theorem that connects a polynomial’s factors and zeros is known as the factor theorem. A polynomial equation can be given by using the factor theorem, which eliminates all known zeros while leaving all unknown zeros. The resulting polynomial has a reduced degree, making it simple to locate the zeros.

According to the factor theorem, if ‘a’ is any real number and f(x) is a polynomial of degree n greater than or equal to 1, then (x – a) is a factor of f(x) if f(a) = 0. Put differently, if f(a) = 0, then (x – a) is a factor of f(x). Now, let’s clarify the meaning of a few terms associated with the factor theorem.

Factor theorem is an essential part of the list of 10th class important theorem.

### 8. Basic Proportionality Theorem

The Basic Proportionality Theorem (BPT), also known as Thales’ Theorem, is a fundamental concept in geometry. It states that if a line is drawn parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally.

Here’s a more formal statement of the theorem:

Suppose we have a triangle ABC, and DE is a line parallel to side BC, with D on side AB and E on side AC.

The Basic Proportionality Theorem states that:

In other words, the ratio of the lengths of the segments of side AB (AD and DB) is equal to the ratio of the lengths of the segments of side AC (AE and EC).

This theorem is particularly useful when working with parallel lines and triangles. It helps establish relationships between the lengths of different segments in a triangle when a parallel line is introduced. Thales’ Theorem is a special case of the more general concept of similar triangles, and it is widely applied in geometry and related mathematical areas.

### 9. Baye’s Theorem

Bayes’ Theorem is a fundamental concept in probability theory that relates the conditional and marginal probabilities of random events. It’s named after the Reverend Thomas Bayes, an 18th-century statistician and theologian. The theorem is expressed as follows:

P(A/B) =  P(B/A). P(A) / P(B)

Where A and B are events and P(B) 0

Here’s what each term represents:

• P(AB): This is the conditional probability of event A occurring given that event B has occurred. It represents the probability of A under the condition that B is true.
• P(BA): This is the conditional probability of event B occurring given that event A has occurred. It represents the probability of B under the condition that A is true.
• P(A): This is the marginal probability of event A. It represents the probability of event A occurring without any condition.
• P(B): This is the marginal probability of event B. It represents the probability of event B occurring without any condition.

The formula essentially tells us how to update our beliefs about the probability of an event (A) given new evidence (B). It’s particularly useful in situations where we want to revise our initial beliefs in light of additional information.

Here’s a common interpretation of the terms:

• P(A) is the prior probability of event A, representing our initial belief in the probability of A.
• P(BA) is the likelihood of observing evidence B given that A is true. It represents how well our initial belief explains the evidence.
• P(B) is the probability of observing evidence B, irrespective of whether A is true.
• P(AB) is the updated probability of A given the new evidence B, which is known as the posterior probability.

Bayes’ Theorem is widely used in various fields, including statistics, machine learning, and artificial intelligence, and it plays a crucial role in Bayesian inference.

## Proof of Math Theorems

Let’s examine Pythagoras’ Theorem in order to demonstrate a theorem.

Theorem of Pythagoras

In a right-angled triangle, the hypotenuse square is equal to the sum of the squares of the other two sides.

The statement that is provided will be used as a guide to first draw a diagram of a right-angled triangle.

Structure to Prove a Theorem:

What details are included in the provided question?

What has to be proven

Building (if necessary)

Method of Proof

As given: The triangle ABC is right-angled.

To demonstrate that AB2 + BC2 = AC2

Draw BD perpendicular to AC in the construction.

Proof:

* ADB ~ ADB If a perpendicular forms from the right angle vertex to the hypotenuse, the triangles on either side of a right triangle are comparable to each other and to the entire triangle.

[Eq.1] states that AB2 = AD × AC because sides AD and AC are proportionate.

* ADB ~ ADB If a perpendicular forms from the right angle vertex to the hypotenuse, the triangles on either side of a right triangle are comparable to each other and to the entire triangle.

As sides CD and AC are proportional, so BC2 = CD × AC —[Eq.2]

Adding equations (1) and (2), we get

AB2 + BC2 = AD × AC + CD × AC

= AC × AC

= AC2

⇒ AB2 + BC2 = AC2

As a result, it demonstrates that the square of the right-angled triangle’s hypotenuse is equal to the sum of the squares of its other two sides.

## How to Study and Learn Math Theorems Easily?

1. Create a notepad dedicated to theorems:

Establish a dedicated notepad for theorems to organize and consolidate your study material efficiently. Ensure that the notepad includes important theorems in mathematics class 10 by referring to NCERT books.

1. Comprehend each theorem:

Ensure a thorough understanding of each theorem, clarifying any uncertainties through active participation in class discussions.

1. Recognise and commit to memory the theorem statements:

Ensure a thorough understanding of each theorem, clarifying any uncertainties through active participation in class discussions. Memorize theorem statements to facilitate quick recall during exams and problem-solving.

1. Every day, practise theorems:

Practice theorems daily to reinforce understanding and develop a strong grasp of the concepts.

1. Create flow diagrams:

Utilize flow diagrams as visual aids to enhance your comprehension of theorem proofs.

1. Allow enough time to yourself to get the theorems ready:

Allocate sufficient time regularly for theorem review and preparation to avoid last-minute cramming.

1. Don’t save any theorem for the last minute:

Ensure to prepare and remember the important theorems in Mathematics Class 10. Avoid procrastination by distributing your theorem study evenly over time and not leaving any theorem for the last minute.

1. Take an exam on yourself to test yourself:

Conduct self-assessment exams periodically to gauge your understanding and identify areas for improvement in mastering the theorems.

The blog covered the most important theorems in mathematics as stated in the NCERT syllabus for Class 10. Continue working at it, never give up, and make an effort to relax. You are the only one who can excel; nobody else will. Believe in yourself.

Also Read: How to Study and Learn Maths Easily

## FAQs on List of Most Important Theorems in Mathematics Class 10

#### Q1. What are the important theorems in mathematics class 10 2023?

Ans. For Class 10 2023, some of the most important theorems are:
• Pythagoras Theorem
• Midpoint Theorem
• Remainder Theorem
• Fundamental Theorem of Arithmetic
• Angle Bisector Theorem
• Inscribed Angle Theorem
• Ceva’s Theorem
• Bayes’ Theorem
Apart from these theorems, the lessons with important theorems are circles and triangles.

#### Q2. Which is the important theorem of triangle class 10 CBSE 2023-24?

Ans. It is difficult to state one theorem as the most important theorem of the triangle. The following theorems below are the basic and yet the most important ones for CBSE 2023-24.
• Theorem 1: The sum of all the three interior angles of a triangle is 180 degrees.
• Theorem 2: The base angles of an isosceles triangle are congruent.
• Theorem 3: The measure of the exterior angle of a triangle is equal to the sum of the corresponding interior angles.

#### Q3. What is the importance of remembering the theorems of class 10 maths?

Ans. Theorems assist students not just in exams but also in preparing for competitive exams and excelling in advanced classes. A key component of the curriculum for competitive exams is the study of theorems. Therefore, the students must put focus on understanding the theorems and getting the foundations right.

#### Q4. What is the importance of theorems in mathematics?

Ans. Theorems are important concepts that are regarded as unchangeable facts. Their proofs not only facilitate the easy solution of mathematical puzzles but also foster a deeper comprehension of the underlying ideas. When students fully comprehend the statements and their proofs, theorems serve as the cornerstone of fundamental mathematics and aid in the development of deductive reasoning.

#### Q5. What is the Pythagoras theorem of triangles class 10?

Ans. Pythagoras’ triangle theorem is arguably the most well-known and frequently discussed.
According to Pythagoras’ Theorem, the square of the hypotenuse side of a right-angled triangle is equal to the sum of the squares of the other two sides. This theorem gives the names perpendicular, hypotenuse, and base to a triangle’s sides. The longest of these three sides, the hypotenuse, is positioned 90 degrees away from the right angle.