Triangles — Mathematics Class 10 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Triangles (Class 10) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Triangles (CBSE & HBSE)
Triangles is a geometry chapter with high marks weightage. CBSE tests similarity theorems and proofs. HBSE focuses on BPT, criteria of similarity, and area ratio theorem.
Similar Triangles and Criteria
Similar Figures
Two figures are similar if:
- Their corresponding angles are equal
- Their corresponding sides are in the same ratio (proportion)
Symbol: △ABC ~ △DEF (read as 'triangle ABC is similar to triangle DEF')
Similar vs Congruent: - Congruent: Same shape AND same size - Similar: Same shape, different sizes (scale copies)
Criteria for Similarity of Triangles
| Criterion | Statement | What to Check |
|---|---|---|
| AA (Angle-Angle) | If 2 angles of one △ = 2 angles of another | Just 2 pairs of equal angles |
| SSS | Ratio of all 3 sides equal | AB/DE = BC/EF = CA/FD |
| SAS | Two sides in ratio AND included angle equal | AB/DE = AC/DF and ∠A = ∠D |
AA is the most commonly used criterion. If two angles match, the third automatically matches (angle sum = 180°).
Basic Proportionality Theorem (BPT) — Thales' Theorem
Statement: If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.
In △ABC, if DE || BC where D is on AB and E is on AC: $$\frac{AD}{DB} = \frac{AE}{EC}$$
Converse of BPT: If a line divides two sides of a triangle in the same ratio, it is parallel to the third side.
Proof of BPT (Important for Exams)
Join BE and CD. Draw DP ⊥ AC and EQ ⊥ AB.
- Area(△ADE)/Area(△DBE) = AD/DB (same height EQ)
- Area(△ADE)/Area(△DEC) = AE/EC (same height DP)
- Area(△DBE) = Area(△DEC) (same base DE, between || lines)
- Therefore: AD/DB = AE/EC ∎
CBSE Pattern: BPT proof is frequently asked as a 5-mark question. Learn it well!
Pythagoras Theorem and Applications
Pythagoras Theorem
Statement: In a right-angled triangle, the square of the hypotenuse equals the sum of squares of the other two sides.
In △ABC, right-angled at B: AC² = AB² + BC²
Proof (Using Similar Triangles): Draw BD ⊥ AC.
- In △ABD and △ABC: ∠A = ∠A, ∠ADB = ∠ABC = 90° → △ABD ~ △ABC
- AB/AC = AD/AB → AB² = AC × AD ...(1)
- In △BCD and △ABC: △BCD ~ △ABC
- BC/AC = DC/BC → BC² = AC × DC ...(2)
- Add (1)+(2): AB² + BC² = AC(AD+DC) = AC × AC = AC² ∎
Converse of Pythagoras Theorem
If in △ABC, AB² + BC² = AC², then ∠B = 90°.
Common Pythagorean Triplets
| Triplet | Check |
|---|---|
| 3, 4, 5 | 9+16=25 ✓ |
| 5, 12, 13 | 25+144=169 ✓ |
| 8, 15, 17 | 64+225=289 ✓ |
| 7, 24, 25 | 49+576=625 ✓ |
| Multiples of above | 6,8,10 or 9,12,15 etc. |
Area Ratio Theorem
Statement: Ratio of areas of two similar triangles = ratio of squares of corresponding sides.
If △ABC ~ △DEF: $$\frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle DEF)} = \frac{AB^2}{DE^2} = \frac{BC^2}{EF^2} = \frac{CA^2}{FD^2}$$
Also equal to: Square of ratio of altitudes, medians, angle bisectors.
Applications of Pythagoras
- Diagonal of rectangle: d = √(l² + b²)
- Distance from ladder base to wall: √(ladder² - height²)
- Diagonal of square with side a: d = a√2
Proof-Based Questions on Similarity
Standard Proofs in Triangles
Theorem 1: BPT (Basic Proportionality Theorem) Statement + Proof (see Module 1)
Theorem 2: AA Similarity If two angles of one triangle equal two angles of another, the triangles are similar.
Theorem 3: Pythagoras Theorem Statement + Proof using similar triangles (see Module 2)
Approach for Proof Questions
- Identify which theorem/criterion to apply
- Establish the required pairs of equal angles or proportional sides
- State the similarity criterion (AA/SAS/SSS)
- Conclude the required result
Common Proof Patterns
Pattern 1: DE || BC → use BPT → AD/DB = AE/EC
Pattern 2: Two triangles share an angle → look for another angle to use AA
Pattern 3: Vertical angles (X-shape) → ∠1 = ∠2 (vertically opposite) → use with another pair for AA
Pattern 4: Parallel lines → alternate angles, corresponding angles (with transversal)
Pattern 5: Right angles given → use Pythagoras or right-triangle similarity
Similarity Shortcut Summary
| Shape | Key Property |
|---|---|
| Equilateral triangles | Always similar |
| Isosceles triangles | Similar only if base angles equal |
| Right triangles | Similar if one acute angle is equal |
CBSE Exam: Proof questions for BPT and Pythagoras Theorem are 5 marks each. Practice writing them with all steps and reasons.
Frequently asked questions
Are these Triangles notes free?
Yes — the Triangles notes for Mathematics (Class 10) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Triangles notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Triangles chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Triangles.