In the figure below, ∠dec ≅ ∠dce, ∠b ≅ ∠f, and segment df is congruent to segment bd. 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.
Angle B and angle F are known to be equal.
Since angles DCE and DEC are equal, angles ACB and GEF are also equal (vertically opposite angles)
When two angles are equal, the third angle has to be equal too because total angle of both triangles is the same, i.e 180
Now when angles are equal, if we just show that lengths BC ans EF are also equal, it would be sufficient to say that the two triangles are congruent.
Since triangle DCE is an isosceles triangle (because two angles same),
Length CD = ED
We know that
Length of BD = Length FD
Length CD = Length ED
Length of BC = BD - CD
Length of EF = FD - ED
Length BC = Length EF
When all angles are equal, and even one pair of corresponding lengths is equal.. the triangles are congruent
∠ABC ≅ ∠DEC
∠DEC = ∠GEF
∠ABC = ∠GEF
∠GFE ≅ ∠DCE
∠DCE = ∠ACB
∠ACB = ∠GFE
If two of their angles are equal, then the third angle must also be equal, because angles of a triangle always add to make 180°.
thus: ΔABC ∼ ΔGEF by AA similar triangle postulate
I hope this helps