Use the isosceles triangle ABC with D on AB and A,E,F,C in order on AC; AD=AE, G=DF∩BE and ∠AFD=∠DEB. Prove △DEG∼△DFE.

01 / Standard#Your turn

Official paper · jm01-2025 · II.4(a) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the given angle and the collinear rays at E and F.

Hint 2

The angle at D is shared.

Worked solution

E,G,B are collinear, and F,E,A lie on one ray from F.
$\angle DEG=\angle DEB=\angle AFD=\angle DFE$
D,G,F lie on one ray from D, giving the second equal angle.
$\angle EDG=\angle FDE\implies\triangle DEG\sim\triangle DFE\quad(AA)$

△DEG∼△DFE by AA.

Checks and common pitfalls: Preserve the correspondence D↔D, E↔F, G↔E when using ratios.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
E,G,B are collinear, and F,E,A lie on one ray from F.
$\angle DEG=\angle DEB=\angle AFD=\angle DFE$
D,G,F lie on one ray from D, giving the second equal angle.
$\angle EDG=\angle FDE\implies\triangle DEG\sim\triangle DFE\quad(AA)$

Think first. Reveal a hint when the class is ready.

2025 JM01