State and prove Pythagoras theorem.

1. 1 · Write the theorem In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
2. 2 · Write what is given and what is to be proved Given: ΔABC with ∠A = 90°. To prove: BC² = AB² + AC². Construction: draw AD ⊥ BC.
3. 3 · Draw the figure Right triangle ABC with the right angle at A, and AD perpendicular to BC. Mark the right angles at A and D.
4. 4 · First pair of similar triangles
   • ΔABC ~ ΔDBA (∠B common, ∠BAC = ∠BDA = 90°)
   • AB/BD = BC/AB, so AB² = BC × BD ... (1)
5. 5 · Second pair of similar triangles
   • ΔABC ~ ΔDAC (∠C common, ∠BAC = ∠ADC = 90°)
   • BC/AC = AC/DC, so AC² = BC × DC ... (2)
6. 6 · Add the results (1) + (2): AB² + AC² = BC × (BD + DC) = BC × BC.
7. 7 · Check and box the answer BC² = AB² + AC². Hence proved.

Scoring tip: Number the two results (1) and (2) so the adding step is clear. Name each pair of similar triangles in matching order.