Important Maths Theorems – Detailed Explanation & Proofs | Testbook

Mathematical theorems are integral to the field of mathematics, with every branch boasting a plethora of established principles put forth by renowned mathematicians worldwide. This article provides a comprehensive list of crucial mathematical theorems for students between classes 6 to 12. These theorems are fundamental in laying a strong foundation in basic mathematics.

A mathematical statement can only be considered a theorem if it has been proven. The act of proving these theorems not only confirms their truth but also aids in the development of logical thinking and reasoning skills, thereby minimizing mathematical errors.

List of Maths Theorems

Pythagoras Theorem	Factor Theorem
Mean Value Theorem	Exterior Angle Theorem
Mid Point Theorem	Cayley Hamilton Theorem
Remainder Theorem	Central Limit Theorem
Isosceles Triangle Theorems	Basic Proportionality Theorem
Greens Theorem	Bayes Theorem
Angle Bisector Theorem	Quadrilateral Theorem
Binomial Theorem	Stewart’s Theorem
Ceva’s Theorem	Apollonius Theorem
Fundamental Theorem Of Arithmetic	Fundamental Theorem of Calculus

A mathematical theorem is a statement that has been proven to be true using logic, known facts, and mathematical rules. We don’t just assume it’s true — there is always a proof that explains why the statement is correct. This proof shows step-by-step reasoning based on things we already know are true.

Why are Theorems Important in Maths

Theorems are very important in mathematics because they are proven facts that we know are always true. They don’t just help us solve problems, but also help students understand the “why” behind mathematical ideas. When students learn how to prove a theorem, they start to think more clearly and logically. Understanding theorems and their proofs builds a strong base in math and helps students develop reasoning and thinking skills that are useful in many areas of learning.

Maths Theorems for Class 10

In Class 10, students learn many important theorems that form the basics of many math topics. These theorems are not just important for exams, but also help build a strong understanding of mathematics. To do well in the board exams, students should make sure to learn each theorem properly, including its statement and proof. Below is a list of some of the key theorems that Class 10 students need to know.

List of Important Class 10 Maths Theorems

In Class 10, there are several important theorems that students should know well. Some of the most commonly used theorems are:

• Pythagoras Theorem
• Midpoint Theorem
• Remainder Theorem
• Fundamental Theorem of Arithmetic
• Angle Bisector Theorem
• Inscribed Angle Theorem
• Ceva’s Theorem
• Bayes’ Theorem

Along with these, students will also find many important theorems in the chapters on circles and triangles. These topics contain some of the most frequently asked theorems in exams, which are listed below.

Circle Theorems for Class 10

There are several key theorems about circles that students should understand clearly. These theorems often appear in school exams and are important for learning geometry. Here are some of the main ones explained simply:

Theorem 1:
If two chords (straight lines inside the circle) are equal in length, they make equal angles at the center of the circle.

Converse:
If two chords make equal angles at the center, then the chords are equal in length.

Theorem 2:
If you draw a line from the center of the circle that is perpendicular to a chord, it will cut the chord exactly in half (bisect it).

Converse:
If a line bisects a chord and also goes through the center of the circle, it must be perpendicular to the chord.

Theorem 3:
Chords that are equal in length are also the same distance from the center of the circle.

Converse:
If two chords are the same distance from the center, then they must be equal in length.

Theorem 4:
The angle made by an arc at any point on the circle’s edge (circumference) is half the angle made at the center by the same arc.

Theorem 5:
In a cyclic quadrilateral (a four-sided figure inside a circle), the opposite angles add up to 180°. These angles are called supplementary.

Triangle Theorems for Class 10

Understanding Similar and Congruent Figures – Made Simple

• All congruent figures are also similar, because they have the same shape and size.
• But the reverse is not always true — not all similar figures are congruent. Similar figures can have the same shape but different sizes.

Similar Polygons

Two polygons (shapes with straight sides) that have the same number of sides are called similar when:

• Their matching angles are equal, and
• Their matching sides are in the same ratio (i.e., one shape is a scaled version of the other).

Similar Triangles

Triangles are a special case. Two triangles are similar when:

• Their matching angles are equal, and
• Their matching sides are in the same ratio.

This means that the two triangles look the same in shape, but they may be of different sizes.

Proof of Theorems

Theorem 1:

If a line is drawn parallel to one side of a triangle and intersects the other two sides, then the other two sides are divided in the same ratio.

Construction: Let's consider triangle ABC, with DE a line parallel to BC, intersecting AB at D and AC at E, i.e., DE || BC. Join C to D and B to E. Draw EM ⊥ AB and DN ⊥ AC.

We need to prove that AD/DB = AE/EC.

Proof:

The area of triangle ADE = ½ × AD × EM, and the area of triangle BDE = ½ × DB × EM. Similarly, the area of triangle ADE = ½ × AE × DN, and the area of triangle DEC = ½ × EC × DN. Hence, the area of triangle ADE to the area of triangle BDE = ½ × AD × EM to ½ × DB × EM = AD to DB. The area of triangle ADE to the area of triangle DEC = AE to EC. Since triangles DEC and BDE are on the same base, i.e., DE, and between the same parallels, DE and BC, their areas are equal. Hence, AD/DB = AE/EC. This proves the theorem.

Theorem 2:

If a line divides any of the two sides of a triangle in the same ratio, then that line is parallel to the third side.

Let's consider triangle ABC, in which DE divides AC and AB in the same ratio. This implies that AB/DB = AE/EC.

Construction: Draw a line DE’ from point D to E’ at AC such that DE’/BC.

Proof:

To prove: If DE` || BC, then AB/DB = AE`/E`C. According to the theorem, AB/DB = AE/EC. Then, accordingly, E and E` must coincide, proving that DE || BC.

Theorem 3:

In two triangles, if the sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal, and the two triangles are similar. This is also called the SSS (side-side-side) criterion.

Let's consider two triangles ABC and DEF that are drawn such that their corresponding sides are proportional, i.e., AB/DE = AC/DF = BC/EF.

Proof:

To prove: ∠A = ∠D, ∠B = ∠E, and ∠C = ∠F, hence triangle ABC ~ DEF. In triangle DEF, draw a line PQ so that DP = AB and DQ = AC. Since the corresponding sides of the two triangles are equal, DP/PE = DQ/QF = PQ/EF, meaning that ∠P = ∠E and ∠Q = ∠F. We had taken ∠A=∠D, ∠B=∠P, and ∠C=∠Q. Hence, ∠A = ∠D, ∠B = ∠E, and ∠C = ∠F. Therefore, from the AAA criterion, triangle ABC ~ DEF.

Theorem 4:

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

Hypotenuse 2 = Base 2 + Perpendicular 2

Proof:

Let's consider triangle ABC, which is right-angled at B. BD is perpendicular to hypotenuse AC, drawn from vertex B.

To prove: AC 2 = AB 2 + BC 2

In the triangle ABC and ADB, AB/AC = AD/AB, meaning that AB 2 = AC × AD. In triangles ABC and BDC, BC/AC = CD/BC, meaning that BC 2 = AC × CD. When we add these two equations, we get AB 2 + BC 2 = AD × AC + CD × AC = AC (AD + CD) = AC 2 . Hence, proved.

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