This article covers all the Class 10 Maths important theorems chapterwise with statements, proof summaries, and real exam applications. These theorems form the backbone of CBSE board preparations. Use this blog to revise quickly before your board exams and build strong conceptual understanding.
Table of Contents
- Chapter 1: Real Numbers
- Chapter 2: Polynomials
- Chapter 3: Pair of Linear Equations
- Chapter 4: Quadratic Equations
- Chapter 5: Arithmetic Progressions
- Chapter 6: Triangles
- Chapter 7: Coordinate Geometry
- Chapter 8: Introduction to Trigonometry
- Chapter 9: Applications of Trigonometry
- Chapter 10: Circles
- Chapter 11: Constructions
- Chapter 12: Areas Related to Circles
- Chapter 13: Surface Areas and Volumes
- Chapter 14: Statistics
- Chapter 15: Probability
- FAQs
Chapter 1: Real Numbers
Theorem: Euclid’s Division Lemma
Statement: For any integers a and b (a > b > 0), there exist unique integers q and r such that a = bq + r, 0 ≤ r < b.
Application: Used to find HCF of two numbers using Euclid’s algorithm.
Read full chapter solutions
Chapter 2: Polynomials
Theorem: Remainder Theorem & Factor Theorem
Statement: If p(x) is a polynomial divided by (x – a), then the remainder is p(a).
Application: Helps factorize cubic and quadratic polynomials quickly.
Chapter 3: Pair of Linear Equations in Two Variables
No direct theorem, but various methods like graphical, substitution, elimination, and cross-multiplication are important.
Application: Used in real-life based word problems.
Chapter 4: Quadratic Equations
Formula: Quadratic Formula
Statement: x = [-b ± √(b² – 4ac)] / 2a
Application: Solve real-life problems involving area, time, speed, etc.
Chapter 5: Arithmetic Progressions
Formula: nth term: an = a + (n – 1)d
Sum of n terms: Sn = n/2[2a + (n – 1)d]
Application: Used in series, savings, and installment problems.
Chapter 6: Triangles
- BPT (Thales Theorem): A line parallel to one side of triangle dividing other two sides in same ratio.
- Pythagoras Theorem: In right-angled triangle, (hypotenuse)² = (base)² + (height)²
- Converse: If this relation holds, triangle is right-angled.
- AA Similarity Theorem: Two triangles are similar if two angles are equal.
- Areas of Similar Triangles: Ratio of areas = square of corresponding sides.
Chapter 7: Coordinate Geometry
Formulas:
Distance between (x1,y1) and (x2,y2): √[(x2 – x1)² + (y2 – y1)²]
Section formula, midpoint formula, area of triangle using coordinates
Application: Finding distances, midpoints and proving collinearity.
Chapter 8: Introduction to Trigonometry
Trigonometric Ratios:
Sinθ = Perpendicular / Hypotenuse, Cosθ = Base / Hypotenuse, etc.
Identities:
1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ, sin²θ + cos²θ = 1
Application: Simplification and proof-based questions
Chapter 9: Applications of Trigonometry
No theorems but problems based on height and distance using trigonometric ratios.
Application: Calculating height of towers, trees, distances between objects, etc.
Chapter 10: Circles
- Theorem 1: The tangent at any point of a circle is perpendicular to the radius at that point.
- Theorem 2: Lengths of tangents drawn from an external point to a circle are equal.
Chapter 11: Constructions
No theorems, but steps of construction for similar triangles, tangents from points outside a circle are important.
Application: Geometry questions in 4–6 marks section.
Chapter 12: Areas Related to Circles
Formulas: Area = πr², Circumference = 2πr
Area of sector and segment using angle subtended at center.
Application: Problems based on park designs, wheel rotations, etc.
Chapter 13: Surface Areas and Volumes
Formulas: Volume and surface area of cube, cuboid, cylinder, cone, sphere, hemisphere
Application: Real-life 3D shapes, conversion of solids, cost estimation.
Chapter 14: Statistics
Formulas: Mean (direct/assumed/deviation), Median (class formula), Mode
Application: Data handling, graph analysis, report conclusions.
Chapter 15: Probability
Theorem: Probability = (Favorable Outcomes) / (Total Outcomes)
Application: Coins, dice, cards, number-based problems.
FAQs
How many theorems are there in Class 10 Maths?
There are around 10–12 important theorems that are asked frequently in the Class 10 board exam.
Which theorem is most important in triangles?
Basic Proportionality Theorem (BPT) and Pythagoras Theorem are the most important from Chapter 6.
Where can I find all formulas?
Visit this page to get chapter-wise formulas for Class 10 Maths.
Can I score full marks in theorems?
Yes, if you understand the statements and practice proofs or applications regularly, you can easily score full marks.