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Justify triangle congruence

Problem

Below are triangle, A, B, C and triangle, D, E, F. We assume that A, B, equals, D, E, B, C, equals, E, F, and m, angle, B, equals, m, angle, E.
Triangle A B C and Triangle D E F. Angles B and E have congruent signs. Side A B and side E D also have congruent signs. Side B C and side E F also have congruent signs.
Here is a rough outline of a proof that triangle, A, B, C, \cong, triangle, D, E, F:
  1. We can map triangle, A, B, C using a sequence of rigid transformations so that A, prime, equals, D and B, prime, equals, E.
  2. If C, prime and F are on the same side of D, E, with, \overleftrightarrow, on top, then C, prime, equals, F.
  3. If C, prime and F are on opposite sides of D, E, with, \overleftrightarrow, on top, then we reflect triangle, A, prime, B, prime, C, prime across D, E, with, \overleftrightarrow, on top and then C, start superscript, prime, prime, end superscript, equals, F, A, start superscript, prime, prime, end superscript, equals, D and B, start superscript, prime, prime, end superscript, equals, E.
What is the justification that C, prime, equals, F in step 2?
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