Triangles

1. Chapter at a glance

• Two figures are similar if they have the same shape but not necessarily the same size; all congruent figures are similar, but similar figures need not be congruent.
• Two polygons with the same number of sides are similar if their corresponding angles are equal and their corresponding sides are in the same ratio (scale factor).
• Two triangles are similar if their corresponding angles are equal and corresponding sides are proportional.
• Basic Proportionality Theorem (Thales Theorem): A line drawn parallel to one side of a triangle intersecting the other two sides divides them in the same ratio.
• Converse of BPT: A line that divides two sides of a triangle in the same ratio is parallel to the third side.
• Similarity criteria for triangles: AAA (or AA), SSS and SAS.
• Applications include proving similarity of triangles, finding unknown lengths, and solving problems on heights/distances using indirect measurement.

2. Definitions, theorems and results

Definitions (NCERT wording): - Two figures having the same shape (and not necessarily the same size) are called similar figures. - Two polygons of the same number of sides are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (or proportion).

Theorems (exact NCERT statements; all require proof in the exam unless stated otherwise): - Theorem 6.1 (Basic Proportionality Theorem): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. - Theorem 6.2 (Converse of Theorem 6.1): If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. - Theorem 6.3 (AAA similarity criterion): If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar. - Theorem 6.4 (SSS similarity criterion): If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar. - Theorem 6.5 (SAS similarity criterion): If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.

Additional results (from text): - If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar (AA similarity criterion; follows from angle sum property). - Remark on transitivity: If one polygon is similar to another and the second is similar to a third, then the first is similar to the third. - Note (RHS similarity): In two right triangles, if hypotenuse and one side of one triangle are proportional to the hypotenuse and one side of the other, then the triangles are similar (mentioned as usable simplification).

3. Formula sheet

No other algebraic formulas; all relations are ratio equalities under similarity or parallelism.

4. Solved-example patterns

• BPT/Converse application: Given parallel line or equal ratios, prove another ratio or parallelism. Steps: State given, apply Theorem 6.1 or 6.2 directly, then use corresponding angles or algebra to reach conclusion.
• Prove triangles similar using criteria: Identify equal angles (vertically opposite, alternate, corresponding) or proportional sides; match to AAA/AA, SSS or SAS. Steps: Write correspondence of vertices, verify two or three conditions, conclude △ABC ~ △DEF with symbol.
• Find unknown lengths/angles using similarity: Establish similarity first, then set up proportion of corresponding sides and solve. Steps: Prove similarity, write \(\frac{AB}{PQ} = \frac{BC}{QR} = \frac{CA}{RP}\), substitute known values, solve for unknown.
• Trapezium/quadrilateral problems: Join diagonal, apply BPT or similarity in resulting triangles. Steps: Use parallel sides to get equal angles, apply Theorem 6.1 or 6.3.
• Real-life indirect measurement (shadows, heights): Form right triangles sharing an angle; use AA similarity. Steps: Draw figure, identify similar triangles (right angle + common angle), set up side proportion, solve for height/length.

5. Common mistakes and exam pitfalls

• Stating two polygons are similar when only angles are equal or only sides are proportional (both conditions required unless using triangle criteria).
• Wrong vertex correspondence when writing △ABC ~ △DEF (must match equal angles/sides).
• Forgetting to prove the third angle equal in AA criterion or omitting “corresponding angles of similar triangles are equal” when finding angles.
• Applying BPT without confirming the line is parallel to the third side or missing “distinct points” condition.
• Sign/unit errors in ratio problems (e.g., mixing cm and m in shadow problems) or assuming similarity without checking included angle for SAS.
• Missing cases: not considering both possible correspondences or overlooking vertically opposite angles in intersecting-line problems.