Triangles is the highest-weightage geometry chapter in Class 10 (about 6–8 marks), and one theorem proof appears almost every year. Learn the three theorems with their proofs.
Similar triangles
- Same shape, corresponding angles equal, corresponding sides in the same ratio.
- Criteria: AAA (or AA), SSS (sides proportional), SAS (two sides proportional + included angle equal).
- All congruent triangles are similar; the converse is false.
Basic Proportionality Theorem (Thales)
If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides them in the same ratio: DE ∥ BC → AD/DB = AE/EC.
Proof idea: join BE and CD; ar(ADE)/ar(BDE) = AD/DB and ar(ADE)/ar(CDE) = AE/EC; since BDE and CDE have the same base DE and same parallels, their areas are equal → ratios equal.
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Converse: if AD/DB = AE/EC then DE ∥ BC.
Area theorem
Ratio of areas of two similar triangles = square of the ratio of corresponding sides: ar(ABC)/ar(PQR) = (AB/PQ)². Example: sides in ratio 4 : 9 → areas 16 : 81.
Pythagoras theorem
- In a right triangle, hyp² = base² + perp². Proof uses similarity: the altitude from the right angle splits the triangle into two triangles similar to the whole.
- Converse: if a² + b² = c², the angle opposite c is 90°.
- Triplets: 3-4-5, 5-12-13, 8-15-17, 7-24-25
Solved examples
1. In ΔABC, DE ∥ BC, AD = 2, DB = 3, AE = 1.6 → EC = 1.6 × 3/2 = 2.4.
2. A ladder 10 m long reaches 8 m up a wall → foot is √(100 − 64) = 6 m from the wall.
3. Two poles 6 m and 11 m are 12 m apart → distance between tops = √(144 + 25) = 13 m.
Common mistakes
- Writing the similarity in wrong order of vertices (ΔABC ~ ΔPQR means A↔P, B↔Q, C↔R)
- Using the area theorem with the ratio of sides, not its square
- Applying Pythagoras in a non-right triangle
Related: Quadratic Equations Class 10.
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Frequently asked questions
Are these Triangles Class 10 Notes notes enough for the exam?
They cover every formula, definition and question type that appears in school and board papers for this chapter. Use them with the NCERT examples, exercise questions and the last five years' papers to be fully prepared.
How should I revise Triangles Class 10 Notes?
Revise the notes after 1 day, 7 days and 30 days. Solve five questions each time instead of only reading, and keep an error list of the steps you get wrong.
What if I get stuck on a triangles class 10 question?
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