How to lay out the answer. Follow the frame, then write it in your own words.
1 · Write the theorem The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.
2 · Write what is given and what is to be proved Given: AD bisects ∠BAC of ΔABC, with D on BC. To prove: AB/AC = BD/DC.
3 · Draw the figure ΔABC, bisector AD, line CE ∥ AB, and AD extended to E. Mark the equal angles.
4 · Write the construction Through C, draw CE ∥ AB. Extend AD to meet CE at E.
5 · Show CE = AC
∠CEA = ∠BAE (alternate angles)
∠BAE = ∠CAE (AD is the bisector)
So ∠CEA = ∠CAE, and CE = AC (ΔACE is isosceles)
6 · Use similar triangles
∠BAD = ∠CED and ∠ADB = ∠EDC (vertically opposite angles)
ΔABD ~ ΔECD (AA), so AB/CE = BD/DC
7 · Check and box the answer Replace CE by AC: AB/AC = BD/DC. Hence proved.
Scoring tip: Do not skip the construction line. Without CE ∥ AB the proof has no reason, and that step carries marks.
