Master Class 10 Maths Triangles: Free NCERT Book PDF Download

For 5-mark long-answer questions, the focus shifts to proofs and higher-order thinking skills (HOTS). The most important areas are:

• Proofs of Theorems: You must be prepared to prove the Basic Proportionality Theorem, the Pythagoras Theorem, and the theorem stating that the ratio of the areas of two similar triangles equals the square of the ratio of their corresponding sides.

• Complex Problems: Questions that combine similarity concepts with the area theorem or require an extra construction are often asked for 5 marks.

• Application of Pythagoras Theorem: Problems involving proofs based on the Pythagoras theorem in various quadrilaterals or triangles.

According to the CBSE syllabus for 2025-26, two theorems are critical for proof-based questions:

• Basic Proportionality Theorem (BPT) or Thales Theorem: The proof for "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."

• Pythagoras Theorem: The proof for "In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides."

Mastering the proofs for these is crucial as they are frequently asked directly.

This is a common point of confusion. You can decide by carefully analysing the information provided in the question:

• Use SSS (Side-Side-Side) if the lengths or the ratio of all three corresponding sides of two triangles are given.

• Use SAS (Side-Angle-Side) if you are given the ratio of two corresponding sides and the measure of the angle included between them.

• Use AAA (Angle-Angle-Angle) or simply AA, if two corresponding angles of two triangles are shown to be equal. This is the most common criterion used in board exam questions.

This is a fundamental concept often tested in HOTS questions. The area of a triangle is a two-dimensional quantity (calculated using base and height). When two triangles are similar with a side ratio of k, it means both their base and their corresponding height are scaled by the same factor 'k'. Since Area = 1/2 × base × height, the new area's ratio becomes proportional to (k × base) × (k × height), which simplifies to k² × (base × height). Therefore, the area ratio is the square of the side ratio (k²), not just the side ratio itself.

Questions based on BPT can be asked in two main ways:

• Direct Proof: You might be asked to state and prove the theorem directly, which usually carries 3 to 4 marks.

• Numerical Application: More commonly, you will be given a triangle with a line parallel to one side, and you'll need to use the property of proportional sides (AD/DB = AE/EC) to find the length of an unknown segment. These are typically 2 or 3-mark questions.

A very common mistake is to assume triangles are congruent when they are only similar. Congruent triangles are identical in size and shape (all sides and angles are equal). Similar triangles have the same shape but can be different sizes (angles are equal, but sides are in proportion). In an exam, if you incorrectly assume congruence, you might state that corresponding sides are equal (e.g., AB = DE) when they are actually proportional (e.g., AB/DE = BC/EF). This mistake leads to completely wrong calculations, especially in problems involving area ratios or finding side lengths.