Triangles Class 10 Notes (2026-27) — CBSE
Class 10 Maths Chapter 6 notes: similar triangles, the criteria for similarity (AAA, SSS, SAS), and the Basic Proportionality (Thales) Theorem with its converse.
Triangles — Class 10 Maths Notes
Chapter Snapshot
This chapter is about similar triangles — triangles with the same shape but not necessarily the same size. You learn what similarity means, the three criteria (AAA, SSS, SAS) that prove two triangles similar, and the Basic Proportionality Theorem (Thales' theorem), which links a line parallel to one side of a triangle to equal ratios on the other two sides.
Board relevance: a proof-heavy Geometry chapter. Expect a similarity proof and a Basic Proportionality Theorem (BPT) application. Write reasons for each step — geometry proofs are marked on reasoning.
Syllabus note (rationalised): Areas of Similar Triangles and the formal Pythagoras Theorem proof (and converse) have been removed from this chapter. Focus on similarity criteria and BPT.
Key Concepts & Definitions
Congruent figures — same shape and size. Similar figures — same shape, possibly different size.
Similar triangles: two triangles are similar if
1. their corresponding angles are equal, and
2. their corresponding sides are in the same ratio (proportional).
We write △ABC ~ △DEF (order matters — it shows which vertices correspond). Then:
∠A = ∠D, ∠B = ∠E, ∠C = ∠F and AB/DE = BC/EF = CA/FD
All congruent triangles are similar (ratio 1:1), but not all similar triangles are congruent.
Basic Proportionality Theorem (Thales' Theorem)
Statement: If a line is drawn parallel to one side of a triangle intersecting the other two sides at distinct points, then it divides the two sides in the same ratio.
In △ABC, if DE ∥ BC (D on AB, E on AC), then:
AD/DB = AE/EC
Converse of BPT: if a line divides two sides of a triangle in the same ratio, then that line is parallel to the third side. (Used to prove lines parallel.)
Formulas — Criteria for Similarity
Criterion Condition
AAA / AA All (or two) pairs of corresponding angles equal ⇒ triangles similar
SSS All three pairs of corresponding sides in the same ratio ⇒ similar
SAS One pair of angles equal and the two sides including them proportional ⇒ similar
- AA is enough: if two angles match, the third does automatically (angle sum), so AAA = AA in practice.
- Once triangles are proved similar, all corresponding sides are in the same ratio — this is how you find an unknown length.
Worked Examples
Example 1 — BPT to find a length: In △ABC, DE ∥ BC with AD = 1.5 cm, DB = 3 cm, AE = 1 cm. Find EC.
By BPT: AD/DB = AE/EC → 1.5/3 = 1/EC → EC = (1 × 3)/1.5 = 2 cm.
Example 2 — Similarity to find a side: △ABC ~ △PQR with AB = 4 cm, BC = 6 cm, and the corresponding PQ = 6 cm. Find QR.
Corresponding sides in ratio: AB/PQ = BC/QR → 4/6 = 6/QR → QR = (6 × 6)/4 = 9 cm.
Example 3 — Prove similarity: In two triangles, ∠A = ∠P and ∠B = ∠Q. Prove △ABC ~ △PQR.
Given ∠A = ∠P and ∠B = ∠Q, by the AA criterion the triangles are similar. (The third angles are then equal by the angle-sum property.) ∎
Example 4 — Converse of BPT: In △ABC, points D and E lie on AB and AC with AD = 2, DB = 4, AE = 3, EC = 6. Is DE ∥ BC?
AD/DB = 2/4 = 1/2 and AE/EC = 3/6 = 1/2. Since the ratios are equal, by the converse of BPT, DE ∥ BC.
Example 5 — Similar triangles in a real figure: A vertical pole 6 m tall casts a shadow 4 m long on the ground; at the same time a tower casts a shadow 28 m long. Find the tower's height.
The sun's rays make the same angle for both, so the pole-and-shadow triangle is similar to the tower-and-shadow triangle (AA). Therefore height/shadow is the same ratio for both: 6/4 = h/28 → h = (6 × 28)/4 = 42 m. This "shadow" set-up — two similar right triangles sharing the sun's angle — is one of the most common real-life similarity questions.
Example 6 — Overlapping triangles: Diagonals of a trapezium ABCD with AB ∥ DC meet at O. Because AB ∥ DC, triangles AOB and COD are similar (AA, using alternate angles), so AO/OC = BO/OD. This equal-ratio result from a pair of similar triangles formed by intersecting lines is a frequently asked proof.
Important Question Patterns
1. BPT application (2–3 marks): find an unknown segment using AD/DB = AE/EC; or use the converse to prove a line is parallel.
2. Prove similarity (3 marks): identify equal angles or proportional sides and cite AA, SSS, or SAS; write △ABC ~ △… in the correct vertex order.
3. Find a length via similarity (2–3 marks): after proving similarity, set up the ratio of corresponding sides to find an unknown.
4. Prove BPT (3 marks): the standard proof using areas of triangles with the same height (a common theorem-proof question).
5. Real-life/figure problems (3–4 marks): shadows, ladders, and overlapping-triangle figures where you spot a pair of similar triangles.
⚡ Quick Revision
- Similar triangles: equal corresponding angles + proportional corresponding sides. Write ~ in correct vertex order.
- Criteria: AA (two angles equal), SSS (all sides proportional), SAS (one angle equal + including sides proportional).
- Basic Proportionality Theorem (Thales): DE ∥ BC ⇒ AD/DB = AE/EC.
- Converse of BPT: equal ratios ⇒ line is parallel to the third side.
- Congruent ⇒ similar (ratio 1:1), but similar ⇏ congruent.
- Once similar, use the equal ratio of corresponding sides to find unknown lengths.
- Removed from syllabus: areas of similar triangles and the Pythagoras theorem proof.
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