Step  Example  Description  Justification 
1


Start with triangle ABC and
DEF.
Let AB be congruent to
DE.
Let BC be
congruent to EF.
Let ∠ABC be congruent to
∠DEF.

Starting conditions.

2


The claim is that AC ≅ DF,
∠BCA ≅ ∠EFD,
∠CAB ≅ ∠FDE,
and ΔABC ≅ ΔDEF.

Claim

3


If ΔABC is placed on top of
ΔDEF, the point A
is on top of point D. Also,
the line segment AB is on top of DE, and the line segment
BC is on top of DF.
Since point A coincides with D and point C
coincides with point F, then the line
segment AC must coincide with the line segment DF.

Euclid Elements Book 1 Common Notion 4: Things which coincide one with another are equal.

4


Since all sides of ΔABC coincide with all sides
of ΔDEF, ΔABC ≅ ΔDEF. It must also be true
that ∠BCA ≅ ∠EFD and ∠CAB ≅ ∠FDE.
QED.

Euclid Elements Book 1 Common Notion 4: Things which coincide one with another are equal.
