Class 10 Maths Chapter 6 Triangles Formula Sheet

The Class 10 Maths Chapter 6 Triangles formula sheet puts every key result on one page. It covers similar figures, the Basic Proportionality Theorem (Thales Theorem) and its converse, all four similarity criteria, the area ratio rule, and the Pythagoras Theorem with its converse. Built on the 2026-27 CBSE syllabus, it is made for quick revision right before the board exam.

• All core theorems in one place, with a plain-English meaning for each.
• Similarity criteria (AA, SSS, SAS, RHS) and the BPT, with the exact conditions to check.
• Board focus: the area ratio rule, the Pythagoras Theorem, common triples, and the proof questions asked every year.

Watch Triangles Class 10 Maths Explained

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Triangles Formula Sheet: Complete List

The table below lists every named result you need from Triangles. Each row gives the concept and its exact statement in simple words, the same one-page list you get in the PDF.

The chapter rests on two big ideas. First, two triangles are similar when they have the same shape (equal angles, proportional sides), and a line parallel to one side cuts the other two sides in the same ratio. Second, the area ratio of similar triangles is the square of the side ratio, and Pythagoras follows from similarity.

Similar Figures: What the Symbols Mean

Two figures are similar when they have the same shape but not always the same size. The symbol ~ reads as "is similar to". Every congruent pair is similar, but similar figures need not be congruent. For two triangles to be similar you need:

• Equal corresponding angles: each angle in one triangle matches an angle in the other in the same position.
• Proportional corresponding sides: each pair of corresponding sides is in the same ratio k, the scale factor.

For triangles you need not check both conditions at once: the four criteria (AA, SSS, SAS, RHS) each prove similarity from one or two facts. Always match the vertices in the correct order. So ▵ABC ~ ▵DEF means ∠A = ∠D, ∠B = ∠E and ∠C = ∠F. Wrong order makes every ratio you read off wrong.

Basic Proportionality Theorem (Thales Theorem): AD/DB = AE/EC

The Basic Proportionality Theorem (BPT) is the base of this chapter. A line drawn parallel to one side of a triangle divides the other two sides in the same ratio. In ▵ABC, if DE ∥ BC with D on AB and E on AC, then AD/DB = AE/EC. You can also write it as AD/AB = AE/AC.

The converse is just as important: if a line divides two sides in the same ratio, it is parallel to the third side. Use it to prove two lines parallel. The mid-point theorem is BPT with ratio 1:1: if D and E are mid-points of AB and AC, then DE ∥ BC and DE = BC/2. If the line is parallel, set up the equal ratios as a proportion to find missing lengths.

Similarity Criteria: AA, SSS, SAS & RHS

These four rules prove two triangles similar by checking only a few angles or sides. Each one is a shortcut, so you skip verifying all six parts of the triangles.

AA is the most used criterion in board proofs: parallel lines, common angles and vertically opposite angles give equal angle pairs for free. SAS needs the equal angle to be the included angle, between the two proportional sides; otherwise it does not apply. For shadow and height problems, both right triangles share the sun's-ray angle, so AA is fastest.

Area Ratio Rule for Similar Figures

If two triangles are similar with scale factor k, their area ratio equals k², the square of the scale factor. This is one of the most tested results in CBSE board papers.

• General rule: ar(▵ABC) / ar(▵DEF) = (AB/DE)² = (BC/EF)² = (CA/FD)²
• Lengths use k, not k²: corresponding medians, altitudes and angle bisectors are in ratio k; only the areas use the square.
• Quick fact: sides in ratio 2:3 give areas 4:9; sides in ratio 1:2 give areas 1:4.

Board questions give the area ratio and ask for the side ratio, or the reverse, so check which way the question runs before you set up the equation.

Pythagoras Theorem and Its Converse

In a triangle right-angled at B, the square on the hypotenuse AC equals the sum of the squares on AB and BC. The Converse says: if AC² = AB² + BC², then the angle at B is 90°. Use the converse to check whether three given lengths form a right triangle.

Spotting a Pythagorean triple saves the full calculation: if the shorter sides are 5 and 12, the hypotenuse is 13. The hypotenuse is always the longest side and sits opposite the right angle.

Picking the Right Similarity Criterion

Most board proofs on Triangles follow a pattern. Once you know which criterion to use, the proof almost writes itself.

The fastest path is to look for equal angles first. Parallel lines give alternate and co-interior angles. A common vertex gives a shared angle. Vertically opposite angles at a crossing are equal. Once you have two equal angle pairs, AA is done. Switch to SSS or SAS only when the question gives side measurements but no angles.

Using This Formula Sheet for Exam Prep

• Night-before revision: read every row in the complete table and check you can write the BPT ratio, all four similarity criteria and the Pythagoras result without looking.
• During practice: keep the PDF open alongside past papers and look up the correct criterion before writing the proof, rather than guessing mid-solution.
• For "prove that" questions: identify what is given (angles, parallel lines, ratios) and match it to the correct criterion row in the table above, then write the proof in order.

Triangles Weightage in the Board Exam

Triangles sits in the Geometry unit of the CBSE Class 10 Maths syllabus. The table shows where its topics fit among common question types, so you can plan revision time well.

Across recent CBSE papers, Triangles carries about 6 to 10 marks. The most common multi-step format asks you to prove two triangles similar, then find a missing measurement. The BPT and AA criterion are reliable scoring points if you know the exact statements.

Common Mistakes With These Theorems

More Triangles Resources

Use this formula sheet with the other Triangles resources below. Each one covers a different part of the chapter, so they work best together.

NCERT Formula Sheets for Class 10 Maths: All Chapters