Triangles - Quick Notes & Formula Sheet
Class 10 CBSE Mathematics · Chapter 6 · Complete Revision Guide (2026-27)
2026-27 Syllabus Update - Read This First
Formal proofs of the theorems in this chapter - the Basic Proportionality Theorem's derivation, the Pythagoras theorem's proof via similarity, and the similarity-criteria proofs - have been removed from the CBSE 2026-27 board exam. You do NOT need to reproduce these proofs step-by-step in the exam. However, applying these theorems to solve numerical problems, find missing sides/ratios, and answer proof-based questions that use the theorems as tools (not proving the theorems themselves) remains fully examinable and high-weightage.
1. Similar Figures & Similar Triangles
Two figures are similar if they have the same shape but not necessarily the same size. For triangles specifically, two triangles are similar if:
- Their corresponding angles are equal, AND
- Their corresponding sides are in the same ratio (proportional)
Trap: Congruent vs Similar
Congruent triangles are always similar (ratio 1:1), but similar triangles are NOT always congruent - they can differ in size. Don't use these terms interchangeably in proofs; CBSE checks this distinction.
2. Basic Proportionality Theorem (BPT / Thales Theorem)
If a line is drawn parallel to one side of a triangle, intersecting the other two sides at distinct points, the other two sides are divided in the same ratio.
If DE || BC in ▵ABC (D on AB, E on AC): AD/DB = AE/EC
Converse of BPT: If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side.
Trap: Using the wrong ratio pairing
Students often write AD/AB = AE/AC (using the whole side) instead of AD/DB = AE/EC (using only the two segments). Both are actually valid depending on how you use it, but be consistent - mixing them (e.g., AD/DB = AE/AC) is a common error.
3. Criteria for Similarity of Triangles
| Criterion | Condition |
|---|---|
| AA (Angle-Angle) | Two angles of one triangle equal two angles of another |
| SAS (Side-Angle-Side) | One pair of equal angles, with the sides forming that angle in proportion |
| SSS (Side-Side-Side) | All three pairs of corresponding sides in proportion |
Quick check tip
AA is the most frequently used criterion in board exams - if you can spot two equal angles (often via parallel lines or a shared/vertically opposite angle), you usually don't need to check side ratios at all.
Trap: Writing corresponding vertices in the wrong order
When writing "▵ABC ~ ▵DEF", the order of vertices MUST match the correspondence (A↔D, B↔E, C↔F). Writing it in a jumbled order is marked wrong even if the triangles genuinely are similar - this is a scored presentation step.
4. Area of Similar Triangles
The ratio of the areas of two similar triangles equals the square of the ratio of their corresponding sides.
Area(▵ABC) / Area(▵DEF) = (AB/DE)² = (BC/EF)² = (AC/DF)²
Trap: Forgetting to square the ratio
A very common mistake: using the side ratio directly as the area ratio, without squaring it. If sides are in ratio 2:3, areas are in ratio 4:9, not 2:3. 
5. Pythagoras Theorem & Its Converse
In a right triangle: (Hypotenuse)² = (Base)² + (Perpendicular)²
Converse: If the square of the longest side of a triangle equals the sum of the squares of the other two sides, the triangle is a right triangle (with the right angle opposite the longest side).
Quick check tip
Use the converse to quickly test if three given side lengths form a right triangle - just check whether the largest number squared equals the sum of the other two squared. Useful for MCQs asking "is this triangle right-angled?"
6. Full Recap - Every Trap in One Place
| Trap | Correct Understanding |
|---|---|
| Treating congruent and similar as the same thing | Congruent = same size and shape; similar = same shape only |
| Mixing segment and whole-side ratios in BPT | Be consistent: AD/DB=AE/EC, or AD/AB=AE/AC - don't cross them |
| Writing similar triangle vertices in the wrong correspondence order | ▵ABC ~ ▵DEF means A↔D, B↔E, C↔F exactly |
| Using the side ratio directly as the area ratio | Area ratio = (side ratio)² - always square it |
| Spending exam time reproducing full theorem proofs | Proofs are deleted for 2026-27 - focus on applying the theorems instead |
Frequently Asked Questions
What is the Basic Proportionality Theorem?
If a line is drawn parallel to one side of a triangle, intersecting the other two sides at distinct points, it divides those two sides in the same ratio.
What are the three criteria for triangle similarity?
AA (two equal angles), SAS (one equal angle with proportional including sides), and SSS (all three sides proportional).
Are theorem proofs required for CBSE 2026-27 in this chapter?
No - formal proofs of BPT, Pythagoras theorem, and similarity criteria are removed from the board exam. Applying these theorems to solve problems remains fully examinable.
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