Statement 2:
Given the triangle CDE:
ΔCDE is isosceles, where DE ≅ EC, so the opposite angles are congruent too:
∠CDE ≅ ∠DCE
Statement 3:
We have:
Then, by alternate interior angles, we conclude that:
∠CDE ≅ ∠EBA
Statement 4:
We have a similar situation as that of the statement 3, so reason is the same to conclude that:
∠DCE ≅ ∠EAB
Statement 5:
We have a similar situation as that of statement 2, so the reason is the same to conclude that:
∠EAB ≅ ∠EBA
Statement 6:
From the previous situation, we see that ΔABE is isosceles, with congruent sides:
EA ≅ EB
Statement 7:
We know that:
EA ≅ EB
ED ≅ EC
AC ≅ EA + EC
BD ≅ EB + ED
Using the two first congruencies on the third one:
AC ≅ EB + ED
Comparing this to the fourth congruence:
AC ≅ BD
So the sum of congruent segments gives us congruent segments.
Statement 8:
The reflexive property states that a quantity is equal to itself. In particular, for the side DC:
DC ≅ DC
Statement 9:
We already know that:
∠CDE ≅ ∠DCE (Statement 2)
DC ≅ DC (Statement 8)
AC ≅ BD (Statement 7)
So, using the Side-Angle-Side theorem, we conclude that ΔACD ≅ ΔBDC