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Most Class 10 students can use the Basic Proportionality Theorem. Far fewer can write its proof cleanly under exam pressure, and the board paper asks for exactly that. The list of proofs is short and fixed. This post gives you that list, a skeleton for each proof, how marks are split, and a routine to make them automatic.
Which theorems does CBSE ask you to prove in 2026-27?
The CBSE Mathematics curriculum for Class X (2026-27) labels some results "(Prove)" and others "State (without proof)". The ones you must be able to prove are:
Real Numbers: proofs of irrationality of √2, √3 and √5, and algebraic proofs for numbers such as 3 + 2√5.
Triangles: 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 (the Basic Proportionality Theorem, or BPT).
Circles: the tangent at any point of a circle is perpendicular to the radius through the point of contact, and the lengths of tangents drawn from an external point to a circle are equal.
Trigonometry: proof and applications of sin²A + cos²A = 1.
The converse of BPT and the AAA, SSS and SAS criteria are "State (without proof)": you use them but need not prove them.
How are proof marks given?
The 2026-27 Class X Mathematics Standard sample paper has a 5-mark question (Q33) that asks you to prove BPT and then apply it to a trapezium. The marking scheme gives 1 mark for "correct figure, given, to prove, construction" and 2 marks for "correct proof". So a third of the theorem marks come before you write a single equation.
The same paper asks you to prove that 2 + 3√3 is irrational, given that √3 is irrational (3 marks), and that a parallelogram circumscribing a circle is a rhombus (3 marks), which runs on the equal-tangents theorem.
What does each proof look like in outline?
These follow the NCERT chapters on real numbers, triangles and circles (Reprint 2026-27).
Root 2 is irrational. Assume √2 = a/b with a and b coprime integers, b not zero. Square: 2b² = a². So 2 divides a², hence 2 divides a (NCERT Theorem 1.2). Write a = 2c, so b² = 2c², and 2 divides b. Now 2 divides both a and b, which contradicts "coprime". The same pattern works for 3 and 5. NCERT marks the proof of Theorem 1.2 itself as "not from the examination point of view", so you quote it, not prove it.
BPT. In triangle ABC with DE parallel to BC, join BE and CD, draw DM perpendicular to AC and EN perpendicular to AB. Write ar(ADE)/ar(BDE) = AD/DB and ar(ADE)/ar(DEC) = AE/EC. Triangles BDE and DEC are on the same base DE and between the same parallels, so their areas are equal. Hence AD/DB = AE/EC.
Tangent is perpendicular to the radius. Take any point Q on the tangent other than P. Q lies outside the circle, so OQ > OP. Since OP is the shortest distance from O to the line, OP is perpendicular to it.
Tangents from an external point are equal. Join OP, OQ, OR. Angles OQP and ORP are right angles by the previous theorem; OQ = OR (radii) and OP is common, so the triangles are congruent by RHS, giving PQ = PR.
sin²A + cos²A = 1. In right triangle ABC with right angle at B, AB² + BC² = AC². Divide by AC².
Worked example: an algebraic irrationality proof
Prove that 3 + 2√5 is irrational, given that √5 is irrational.
Assume, to the contrary, that 3 + 2√5 is rational. Then 3 + 2√5 = a/b for integers a and b, b not zero. Rearranging, √5 = (a − 3b)/(2b). Since a and b are integers, (a − 3b)/(2b) is rational, so √5 would be rational. This contradicts the given fact. Hence 3 + 2√5 is irrational.
The sample paper's scheme for the √3 version follows the same three steps: assumption, isolating the root, contradiction.
What mistakes cost marks?
Assuming what you need. In the tangent theorem, writing "angle OPX = 90° because it is a tangent" uses the result to prove itself.
Forgetting "coprime". The √2 proof only works if a and b have no common factor. Without that line there is no contradiction.
Using the converse to prove BPT. The converse is a different statement. Keep "if parallel then same ratio" and "if same ratio then parallel" apart.
An unlabelled figure. The construction lines (BE, CD, DM, EN) must appear on it.
Why do students who know the facts still get stuck?
Knowing facts and knowing how to use them in a proof are different. Keith Weber (2001) compared undergraduates and doctoral students writing proofs and found that undergraduates often knew the facts needed but still failed, because they lacked strategic knowledge of which facts and theorems to apply. For Class 10 the fix is simple: for each theorem, learn the one key move (area ratios for BPT, shortest distance for the tangent, RHS for equal tangents, divisibility for √2) and why it is chosen.
If you still cannot see why a step follows, the Fermi Problem Library has a step-by-step solver that explains the reasoning rather than printing an answer, which helps when a textbook proof skips a line.
A practice routine for the proofs
Week 1: Write each proof once with the book open. Underline the key move.
Week 2: Write each from memory, with figure, given, to prove and construction. Check against the book and mark every gap. Testing yourself like this strengthens memory more than rereading; Roediger and Karpicke (2006) found that prior testing gave much better retention on delayed tests than restudying.
Week 3 onward: Twice a week, pick one proof at random and one application question, such as the trapezium in Q33. Read more on the circle results in our post on tangents to a circle in Class 10 and on BPT in similar triangles in Class 10.
Check the current year's CBSE curriculum before your exam, since the list can change.
Tonight, write the BPT proof from memory with a labelled figure, then compare it line by line with your textbook.
