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Schooling · Class 10 · Mathematics · Triangles · Chapter on a page

CBSE Class 10 · Mathematics · NCERT Mathematics

Triangles: chapter on a page

A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).

About the notes: Written by Shishya's AI on 26 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.

This note's current text has no applied fact-check on record.

Mind map

Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.

Mind map of Triangles: Key ideas, Formulas and facts, Watch out forKey ideasTwo polygons are similar if their…Basic Proportionality Theorem…The converse also holdsAAA (or AA) CriterionSSS Criterion for SimilaritySAS Criterion for SimilarityThe ratio of the areas of two…Formulas and factsIf △ABC ~ △DEF with scale factor k…Area(△ABC) / Area(△DEF) = k² =…Basic Proportionality TheoremPythagoras Theorem (special case of…Converse of Pythagoras TheoremWatch out forConfusing congruence with similarityWriting ratios of non-corresponding…Forgetting to square the ratio when…Assuming any two triangles with…Applying the Basic Proportionality…Triangles
The mind map as a list
  • Key ideas
    • Two polygons are similar if their…
    • Basic Proportionality Theorem…
    • The converse also holds
    • AAA (or AA) Criterion
    • SSS Criterion for Similarity
    • SAS Criterion for Similarity
    • The ratio of the areas of two…
  • Formulas and facts
    • If △ABC ~ △DEF with scale factor k…
    • Area(△ABC) / Area(△DEF) = k² =…
    • Basic Proportionality Theorem
    • Pythagoras Theorem (special case of…
    • Converse of Pythagoras Theorem
  • Watch out for
    • Confusing congruence with similarity
    • Writing ratios of non-corresponding…
    • Forgetting to square the ratio when…
    • Assuming any two triangles with…
    • Applying the Basic Proportionality…

What this chapter is about

This chapter explores the concept of similarity of triangles, building upon the idea of congruence studied in earlier classes. While congruent figures are exact copies (same shape and size), similar figures have the same shape but may differ in size. You will learn precise conditions under which two triangles are similar and how their sides and areas relate.

Key ideas

  • Two polygons are similar if their corresponding angles are equal and corresponding sides are in the same ratio (called the scale factor or ratio of similitude).
  • Basic Proportionality Theorem (Thales Theorem): If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.
  • The converse also holds: If a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.
  • AAA (or AA) Criterion: If in two triangles, all three pairs (or any two pairs) of corresponding angles are equal, the triangles are similar.
  • SSS Criterion for Similarity: If the corresponding sides of two triangles are in the same ratio, the triangles are similar.
  • SAS Criterion for Similarity: If one angle of a triangle equals one angle of another, and the sides including these angles are in the same ratio, the triangles are similar.
  • The ratio of the areas of two similar triangles equals the square of the ratio of their corresponding sides.
  • In a right-angled triangle, the altitude drawn from the right angle to the hypotenuse divides the triangle into two triangles, each similar to the original and to each other.

Formulas and facts to remember

  • If △ABC ~ △DEF with scale factor k, then AB/DE = BC/EF = CA/FD = k.
  • Area(△ABC) / Area(△DEF) = k² = (AB/DE)² = (BC/EF)² = (CA/FD)².
  • Basic Proportionality Theorem: If DE ∥ BC in △ABC with D on AB and E on AC, then AD/DB = AE/EC.
  • Pythagoras Theorem (special case of similarity): In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
  • Converse of Pythagoras Theorem: If in a triangle the square of one side equals the sum of the squares of the other two sides, the angle opposite the first side is a right angle.

Common mistakes

  • Confusing congruence with similarity → Congruent triangles are equal in size; similar triangles only have equal angles and proportional sides.
  • Writing ratios of non-corresponding sides → Always match vertices in the same order; if △ABC ~ △DEF, then AB corresponds to DE, not DF.
  • Forgetting to square the ratio when comparing areas → Area ratio is the square of the side ratio, not the same as the side ratio.
  • Assuming any two triangles with proportional sides are congruent → They are similar, but congruent only when the scale factor is 1.
  • Applying the Basic Proportionality Theorem when the line is not parallel to a side → The theorem requires the line to be parallel to one side of the triangle.

Quick revision

  • Similar triangles have equal corresponding angles and proportional corresponding sides.
  • A line parallel to one side of a triangle divides the other two sides proportionally (BPT).
  • Two triangles are similar if two pairs of angles are equal (AA criterion).
  • Ratio of areas of similar triangles = (ratio of corresponding sides)².
  • The altitude to the hypotenuse of a right triangle creates two triangles similar to the original.
  • Pythagoras Theorem follows from similarity of right triangles.