State and prove Thales theorem (basic proportionality theorem).

1. 1 · Write the theorem A straight line drawn parallel to one side of a triangle cuts the other two sides. It divides them in the same ratio.
2. 2 · Write what is given and what is to be proved Given: in ΔABC, DE ∥ BC, with D on AB and E on AC. To prove: AD/DB = AE/EC.
3. 3 · Draw the figure ΔABC with DE parallel to the base BC. Label A, B, C, D, E.
4. 4 · Prove the triangles similar
   • ∠A is common to ΔADE and ΔABC.
   • ∠ADE = ∠ABC (corresponding angles, DE ∥ BC).
   • So ΔADE ~ ΔABC (AA).
5. 5 · Use the ratio of sides
   • AB/AD = AC/AE
   • (AD + DB)/AD = (AE + EC)/AE
   • 1 + DB/AD = 1 + EC/AE, so DB/AD = EC/AE
6. 6 · Check and box the answer Taking reciprocals, AD/DB = AE/EC. Hence proved.

Scoring tip: Give a reason in brackets for every angle you call equal. 'Corresponding angles' and 'AA' are the words examiners look for.