Mathematics: Class 10 — Chapter 6: Triangles

6.1 & 6.2 Introduction to Similarity

Congruence vs. Similarity

Congruent Figures: Geometric figures that have the exact same shape and the same size.
Similar Figures: Geometric figures that have the same shape but not necessarily the same size.
Key Principle: All congruent figures are similar, but similar figures need not be congruent.
Examples of Always Similar Figures: All circles, all squares, and all equilateral triangles.

Similarity of Polygons

Two polygons with the same number of sides are similar if:
1. Their corresponding angles are equal.
2. Their corresponding sides are in the same ratio (proportional).
Scale Factor: The ratio of the corresponding sides of similar polygons (also called Representative Fraction).

6.3 Similarity of Triangles & Fundamental Theorems

Similarity Conditions for Triangles

Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.
Equiangular Triangles: Triangles that have equal corresponding angles. The ratio of any two corresponding sides in equiangular triangles is always the same (Thales' Principle).

Theorem 6.1: Basic Proportionality Theorem (BPT)

Statement: 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.
Mathematical Form: In ΔABC, if DE || BC, then AD/DB = AE/EC.

Theorem 6.2: Converse of BPT

Statement: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
Mathematical Form: In ΔABC, if AD/DB = AE/EC, then DE || BC.

6.4 Criteria for Similarity of Triangles

AAA and AA Similarity Criteria

AAA Similarity Criterion (Theorem 6.3): If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio, and hence the triangles are similar.
AA Similarity Criterion: If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.

SSS Similarity Criterion

SSS Similarity Criterion (Theorem 6.4): If in two triangles, 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.

SAS Similarity Criterion

SAS Similarity Criterion (Theorem 6.5): 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.

RHS Similarity Criterion

RHS Similarity Criterion: If in two right triangles, the hypotenuse and one side of one triangle are proportional to the hypotenuse and one side of the other triangle, then the two triangles are similar.