Triangles Class 10 Maths in One Shot Concepts Examples CBSE Class 10 MidTerm Board Exams

Triangles: Similarity, Congruence, and the Pythagorean Theorem

Chapter 6 of CBSE Class 10 Mathematics deals with the properties of triangles, focusing primarily on the concept of similarity and its powerful applications. While congruent figures are identical in both shape and size (all corresponding sides and angles are equal), similar figures have the same shape but not necessarily the same size — all corresponding angles are equal and all corresponding sides are in the same ratio. This ratio is called the scale factor or similarity ratio. Two triangles are said to be similar (denoted by the symbol ~) if their corresponding angles are equal and their corresponding sides are proportional. Understanding similarity is crucial because it connects geometry with real-world applications such as map-making, architecture, surveying, and photography, where objects are represented at different scales while preserving their proportions.

The chapter establishes three criteria for similarity of triangles. The AAA (Angle-Angle-Angle) criterion states that if all three pairs of corresponding angles of two triangles are equal, then the triangles are similar. Since the sum of angles in any triangle is always 180°, it is sufficient to show that two pairs of angles are equal — this is the AA criterion, which is the most frequently used in practice. The SSS (Side-Side-Side) criterion states that if the corresponding sides of two triangles are in the same ratio, the triangles are similar. The SAS (Side-Angle-Side) criterion states that if one pair of corresponding angles is equal and the sides including those angles are in the same ratio, the triangles are similar. The Basic Proportionality Theorem (BPT), also known as Thales' theorem, is a fundamental result: if a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides those two sides in the same ratio. Its converse is equally important — if a line divides two sides of a triangle in the same ratio, then it is parallel to the third side.

The Pythagorean theorem, which states that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides (a² + b² = c²), is actually a special case of similarity applied to right-angled triangles. By dropping an altitude from the right angle to the hypotenuse, the original triangle is divided into two smaller triangles, each similar to the original triangle and to each other. This decomposition leads directly to the Pythagorean theorem and several important corollaries. The converse of the Pythagorean theorem is also true: if the square of the longest side of a triangle equals the sum of the squares of the other two sides, then the triangle is right-angled. The chapter also covers the area ratio property — the ratio of the areas of two similar triangles equals the square of the ratio of their corresponding sides. This means that if the scale factor between two similar triangles is k, the ratio of their areas is k².

• Similar triangles have equal corresponding angles and proportional corresponding sides; congruent triangles are identical in shape and size.
• AA, SSS, and SAS are the three criteria for establishing similarity of triangles.
• Thales' theorem (BPT): a line parallel to one side of a triangle divides the other two sides proportionally.
• The Pythagorean theorem (a² + b² = c²) and its converse are derived from properties of similar right-angled triangles.
• The ratio of areas of two similar triangles equals the square of the ratio of their corresponding sides.