Class 10 Maths Proof Writing: Reasons and Worked Example

Build the logical chain

1. 1

   Separate given and required

   Write exactly what is known and exactly what must be shown.

2. 2

   Choose a bridge

   Identify a theorem, definition or algebraic property connecting them.

3. 3

   Justify each new statement

   Use the given facts or an earlier established statement, not the desired conclusion.

4. 4

   Close the argument

   Name the conclusion and explain why your chain establishes it.

A short irrationality argument

Suppose, for contradiction, that 4 + 3√2 is rational, and call it r. Subtracting 4 and dividing by the nonzero rational number 3 would make √2 = (r − 4)/3 rational. This contradicts the known irrationality of √2. Therefore 4 + 3√2 is irrational.

The conclusion uses closure of rational numbers under subtraction and division by a nonzero rational number. It does not use the false rule that every sum of irrational numbers is irrational. For instance, √2 + (−√2) = 0.

Make a geometry proof readable

State the pair of triangles in corresponding order. Beside an angle equality, give its reason: vertically opposite angles, corresponding angles from parallel lines, or a stated condition. Then name the similarity or congruence criterion before extracting the relation you need.

A diagram may be rotated or drawn out of scale. Avoid adding a right angle or equal side just because it appears that way. Extra construction lines are useful when you explain what they join and how they supply a missing relation.

Repair the exact missing inference

Flawed argument: ‘AB/DE = AC/DF, so triangle ABC is similar to triangle DEF.’ Two proportional sides alone do not prove similarity. The included angles must also be equal for SAS similarity; an unrelated angle equality is insufficient.

Repair under the extra given ∠BAC = ∠EDF: state AB/DE = AC/DF and equality of the included angles, then conclude ΔABC ∼ ΔDEF by SAS. Only now infer BC/EF = AB/DE. If that included-angle equality is not given or established, this repair is unavailable.

Repair the argument

Swipe across the table to see all columns.

Use the marking scheme after an attempt

Compare your chain with the official scheme for that exact paper. A different valid method can still be mathematically sound; inspect whether you established the necessary relationships. Do not assume this general writing guide promises a fixed mark for every line.

Choose what to study next

Sources and scope

This is a focused learning resource, not a complete chapter or a prediction of board questions. Use your current syllabus and the paper-specific marking scheme for assessment requirements.