Did You Study The Most Important Math Theorems As Per NCERT Class 10 Syllabus?

Math is an essential and fundamental subject in the educational curriculum. It aids in the intellectual growth of students. Students also learn mathematical theorems, which stimulate their brain activity. However, math can become the most challenging subject to understand, the most monotonous subject at times, or both.

Proof of theorems is a common question repeatedly posed in board exams, according to our analysis of the question types in the NCERT Class 10 syllabus for Math. Although these mathematical theorems can be challenging to understand at first, they make up most of the marks in the paper. Thus, students should prepare for mathematics well during study and practice.

So, let's go through some of the essential math theorems as per NCERT Class 10 syllabus. When preparing for the exam, you must understand the process for proving these theorems and practice using them.

List of Important Theorems for NCERT Class 10

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

Other than these theorems, real numbers, circles, and triangles are the lessons with the essential theorems. The following list includes several fundamental math theorems in NCERT Class 10 syllabus.

Real Numbers

Real numbers are only the fusion of rational and irrational numbers in the number system. These numbers are generally used for all arithmetic operations and expressed on a number line. So, let’s learn about the proof of irrationality theorem.

• Proof of Irrationality of √2

Consider√2 is an irrational number.

√2 = a/b (as a and b are co-prime)

⇒√2 b = a

⇒2b² = a²

⇒ b² = a²/b

So dividing a² by 2, 2 also divides a. Consider a = 2c.

⇒ a² = 4c²

⇒2b² = 4c² (as a² = 2b²)

⇒b² = 2c²

⇒c² = b²/b

As 2 divides b², so 2 also divides b. So, a and b have 2 at least as a common factor, but earlier, it stated that a and b are co-prime (no common factor).

Thus, it contradicts our assumption that√2 is rational. So,√2 is irrational.

• Proof of Irrationality of√5

Consider√5 is a rational number.

So√5 = p/q (as p and q are co-prime in nature)

On squaring, 5 = p²/q² (Equation 1)

⇒ q² * 5 = p²

Here, 5 divides p², and then 5 divides p.

Consider, p = 5a

Summing up in equation 1, q² * 5 = (5a)²

⇒ q² = 5 (a)²

5 divides q, so 5 divides p and q both.

But this is false as p and q are co-prime, so here our assumption is wrong.

So,√5 is an irrational number.

• Circles

There are many theorems connected to the circle. Both Class 9 and Class 10 students should be familiar with the circle theorems. Several essential theorems include:

Theorem 1: At the circle's center, equal chords subtend equal angles.

The converse of Theorem 1: If two angles that two chords subtend at their centers are equal, then the length of the chords is also equal.

Theorem 2: If drawn from the center of the circle, then the perpendicular to a chord bisects the chord. The converse of Theorem 2: A straight line bisecting a chord across the center of the circle is hence, perpendicular to the chord.

Theorem 3: Equidistant (equal distance) from the circle's center is equal chords of a circle.

The converse of Theorem 3: A circle's chords have equal lengths when located at equal distances from the center.

Theorem 4: The angle that the same arc subtends at any point along the circumference of the circle is equal to half of the angle it subtends at the center.

Theorem 5: In a cyclic quadrilateral, the opposite angles are supplementary.

• Triangles

• Even while all congruent figures are similar, not all comparable figures are congruent.
• If the corresponding angles and sides of two polygons with the same number of sides hence are identical, then the polygons are comparable.
• If the corresponding angles and sides of two triangles are identical, then the triangles are comparable.
• When the respective sides of two triangles have the same ratio, and the corresponding angles are equal, then such triangles are said to be as similar.
• When the sides of any two triangles are in proportion to one another, their corresponding angles are also in proportion, and the two triangles are said to be similar.

How to Prove a Theorem?

To prove a theorem, let us consider a theorem- Pythagoras’ Theorem.

• Pythagoras' Theorem

The hypotenuse square in a right-angled triangle equals the sum total of the squares of the other two sides.

We'll first draw a right-angled triangle diagram using the provided statement as a reference.

Next, we'll put it in writing.

• What information is there in the question given
• What we must prove
• Construction (if required)
• Way of proving

Provided: ABC triangle is a right-angled triangle.

To prove that AC² = AB² + BC²

Construction: Draw BD perpendicular to AC.

• Proof

ADB ~ ADB (Right triangles have triangles on both sides that are comparable to the entire triangle and one another if a perpendicular forms from the right angle vertex to the hypotenuse.)

As sides AD and AC are proportional, so AB2 = AD × AC —[Eq.1]

ADB ~ ADB (Right triangles have triangles on both sides that are comparable to the entire triangle and one another if a perpendicular forms from the right angle vertex to the hypotenuse.)

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 (AD + CD)

= AC × AC

= AC²

⇒ AB² + BC² = AC²

Thus, it shows that the square of the hypotenuse in the right-angled triangle equals the sum total of the squares of the other two sides.

Math Theorem Preparation: Do’s and Don’t’s

Guidelines for Practicing Math Theorems

• Make a theorems-only notebook.
• You must understand each theorem thoroughly.
• If you need help understanding any of the steps in the proof, ask questions in class.
• Understand and memorize the theorem statements.
• Practice theorems every day.
• Make flowcharts.
• Try explaining the proof to your friends.
• Give yourself adequate time to prepare the theorems.
• Leave no theorem until the very last minute.
• Test yourself by taking your own test.

Cons of Memorizing Math Theorems

• You can entirely forget about them.
• You might not be able to understand the theorem thoroughly.
• At the moment, you might fail to get the relationship between the old and new knowledge.
• You might become distracted.
• You won't be able to learn how to solve problems.

Conclusion

So that's it for the most crucial mathematical theorems according to the NCERT Class 10 syllabus. Never give up, keep working at it, and try not to worry. You must excel yourself since no one else will. Have faith in yourself. Your enthusiasm for learning math is sure to soar! And if you need any assistance, feel free to contact Physics Wallah.

Frequently Asked Questions (FAQs)

Q.1. What Do Math Theorems Mean?

Ans. Statements proven true by preceding accepted claims, mathematical operations, or reasoning are known as mathematical theorems. There is a proven proof that supports the integrity of each mathematical theorem.

Q.2. Why are Math Theorems Important?

Ans. Theorems have importance and are unchanging realities. Theorems not only make it easier to solve mathematical problems, but their proofs also help us comprehend the underlying ideas more fully. When students fully understand the statements and their justifications, theorems serve as the cornerstone of fundamental mathematics and aid in developing deductive thinking.

Q.3. Is it OK to memorize Math theorems while preparing for Class 10 Boards?

Ans. No, it's not a good idea to memorize Math theorems while preparing for class 10 board exams. It is easier to get higher marks in mathematics because the subject is mostly numbers. To perform better, we advise that students first understand and learn the theorems.

So, try to understand and solve the theorems correctly by following the steps along with genuine reasons to help you know the theorems thoroughly. And, if you feel you have any doubts, get your queries resolved by your concerned teacher.