Consider two triangles △ABC and △DEF, whose two pairs of corresponding sides are and the included angles are congruent.
These triangles can be proven to be similar by identifying a that maps one triangle onto the other. First, △DEF can be with the k=DEAB about D, forming the new triangle △DE′F′.
Because dilation is a similarity transformation, it can be concluded that
△DE′F′ and
△DEF are . Now, it has to be proven that a that maps
△DE′F′ onto
△ABC exists. The ratios of the corresponding side lengths of are the same and equal to the scale factor.
DEDE′=DFDF′=k
In this case, the scale factor
k is
DEAB. Since
AB and
AC are proportional to
DE and
DF respectively, the scale factor can be expressed by any of the following ratios.
k=DEAB=DFAC
Applying the , three equations can be formed and simplified.
DEDE′=DEABDFDF′=DFAC⇒⇒DE′=ABDF′=AC
These relations imply that the two sides of
△DE′F′ are to the corresponding two sides of
△ABC. Moreover, the included angles
∠A and
∠D are also congruent.
Therefore, by the , the two triangles are congruent.
△ABC≅△DE′F′
Since congruent figures can be transformed into each other using rigid motions, and
△ABC and
△DE′F′ are congruent triangles, there is a rigid motion placing
△DE′F′ onto
△ABC.
The combination of this rigid motion and the dilation performed earlier forms a similarity transformation that maps △DEF onto △ABC.
Therefore, it can be concluded that △ABC and △DEF are similar triangles.
The proof is now complete.