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 △DEF∼△BDE.

01 / Standard#Your turn

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

AD=AE makes triangle ADE isosceles.

Hint 2

Use supplementary angles along AB and AC.

Worked solution

Equal base angles in ADE give equal supplementary angles.
$\angle ADE=\angle AED\implies\angle BDE=\angle DEF$
The given angle gives the other equal pair.
$\angle DFE=\angle AFD=\angle DEB\implies\triangle DEF\sim\triangle BDE$

△DEF∼△BDE by AA.

Checks and common pitfalls: The exterior angles here are supplements of the equal base angles.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Equal base angles in ADE give equal supplementary angles.
$\angle ADE=\angle AED\implies\angle BDE=\angle DEF$
The given angle gives the other equal pair.
$\angle DFE=\angle AFD=\angle DEB\implies\triangle DEF\sim\triangle BDE$

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

2025 JM01