Question

In the figure below, ∠DEC ≅ DCE, ∠B ≅ ∠F, and segment DF is congruent to segment DB. Point C is the point of intersection between segment AG and segment BD, while point E is the point of intersection between segment AG and segment DF.

The figure shows a polygon comprised of three triangles, ABC, DEC, and GFE.

Prove ΔABC ≅ ΔGFE.

Answer 1

given ∠DEC ≅ DCE, hence DC=CE aslo given DB=Df
hence
DB- DC =DF -CE
BC =EF
given ∠DEC ≅ DCE
but <DCE = <ACB (vert. opposite angles)
but ∠DEC = <GEF (vert. opposite angles)
hence <ACB = <GEF
now in ΔABC and ΔGFE.
BC =EF (proved earlier)
<ACB = <GEF (proved earlier)
∠B ≅ ∠F (given)
hence ΔABC ≅ ΔGFE. (AAS congruency)