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Ch 6Triangles

Geometry — 15 marks

Chapter 6: Triangles

1. Similar Figures

Two figures having the same shape (but not necessarily the same size) are called similar figures. All congruent figures are similar, but similar figures need not be congruent.

  • All circles are similar.
  • All squares are similar.
  • All equilateral triangles are similar.
  • Two polygons with the same number of sides are similar if (i) their corresponding angles are equal AND (ii) their corresponding sides are in the same ratio (proportion).

2. Similar Triangles

Two triangles are similar if:

  1. Their corresponding angles are equal, and
  2. Their corresponding sides are in the same ratio (proportion).
If ΔABC ~ ΔPQR, then ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R and AB/PQ = BC/QR = AC/PR
The symbol "~" is read as "is similar to". Always write the correspondence in the correct order: ΔABC ~ ΔPQR means A ↔ P, B ↔ Q, C ↔ R. Wrong order = wrong ratios = lost marks.

3. Basic Proportionality Theorem (BPT / Thales Theorem) — Theorem 6.1

Statement: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

In ΔABC, if DE ∥ BC with D on AB and E on AC, then AD/DB = AE/EC

Proof outline (asked in boards — learn fully):

  1. Given: ΔABC in which DE ∥ BC intersects AB at D and AC at E.
  2. To prove: AD/DB = AE/EC.
  3. Construction: Join BE and CD. Draw DM ⊥ AC and EN ⊥ AB.
  4. ar(ΔADE) = ½ × AD × EN and ar(ΔBDE) = ½ × DB × EN, so ar(ΔADE)/ar(ΔBDE) = AD/DB.
  5. Similarly, ar(ΔADE) = ½ × AE × DM and ar(ΔDEC) = ½ × EC × DM, so ar(ΔADE)/ar(ΔDEC) = AE/EC.
  6. ΔBDE and ΔDEC stand on the same base DE and lie between the same parallels DE and BC, so ar(ΔBDE) = ar(ΔDEC).
  7. Therefore AD/DB = AE/EC. Hence proved.

4. Converse of BPT — Theorem 6.2

Statement: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

In ΔABC, if AD/DB = AE/EC, then DE ∥ BC
Only the statement and applications of the converse are required; its proof is not asked. But the proof of BPT itself is a favourite 3-mark/5-mark board question — practise writing it with the figure.

5. Criteria for Similarity of Triangles

(Statements to be learnt; proofs of these criteria are not in the syllabus.)

(a) AAA Similarity Criterion

If in two triangles, the corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar.

(b) AA Similarity Criterion

If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. (The third angles are automatically equal by the angle sum property, so AA is enough — this is the most used criterion in board problems.)

(c) SSS Similarity Criterion

If in two triangles, the sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal and hence the triangles are similar.

(d) SAS Similarity Criterion

If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, then the two triangles are similar.

While proving similarity, always state the criterion used, e.g. "ΔABC ~ ΔPQR (by AA similarity criterion)". Missing the reason costs marks.

6. Important Results Used in Problems

  • If DE ∥ BC in ΔABC, then ΔADE ~ ΔABC (by AA), so AD/AB = AE/AC = DE/BC.
  • Vertically opposite angles and alternate angles (with parallel lines) are the standard tools to get the two equal angles needed for AA.
  • Ratios of corresponding sides of similar triangles equal the ratios of their corresponding medians, altitudes and angle bisectors (use only if proved in the answer or given).
  • A ladder/pole/shadow question: two vertical objects and their shadows at the same time form similar triangles (both make the same sun angle, both stand at 90°) — use AA.

7. Method for Typical Board Questions

  1. Draw a neat labelled figure — almost every question in this chapter requires it.
  2. Identify the pair of triangles to compare; mark equal angles.
  3. Write the similarity statement in correct correspondence with the criterion.
  4. Write the proportionality of sides and substitute values.
  5. Solve the resulting equation; state units in the final answer.

Common board patterns: prove BPT (3/5 marks); find x using DE ∥ BC (1/2 marks); prove two triangles similar and hence find a side (3 marks); shadow/height similarity word problem (2/3 marks); case-study on similarity in real objects such as poles, ramps, pyramids (4/5 marks).

Deleted in rationalised syllabus (2026-27): Proofs of all theorems other than the Basic Proportionality Theorem are deleted — you need only the statements of the similarity criteria and of the converse of BPT. The theorem on the ratio of areas of similar triangles is deleted entirely. The proof of the Pythagoras theorem and the proof of its converse (via similarity) are deleted; the Pythagoras results themselves continue to be used from Class 9 knowledge where needed.
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