Triangles ABC and DEF have a corresponding angle measuring $$ 40^{\circ} $$. What other piece of information is sufficient to determine these triangles are similar?

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1. We know that both triangles have a corresponding angle of $$ 40^{\circ} $$.

2. One way to establish similarity is by the AA (angle–angle) criterion, which requires two corresponding angles to be congruent. Thus, if you know another pair of corresponding angles (in addition to the given $$40^{\circ}$$ angle) are equal, the triangles are similar.

3. Alternatively, you can use the SAS (side–angle–side) similarity criterion. In this case, if you know that the two pairs of sides that enclose the $$ 40^{\circ} $$ angle are in proportion (i.e. if
   $$
   \frac{\text{side}_1 \text{ in } \triangle ABC}{\text{corresponding side}_1 \text{ in } \triangle DEF} = \frac{\text{side}_2 \text{ in } \triangle ABC}{\text{corresponding side}_2 \text{ in } \triangle DEF}
   $$
   where these sides meet at the $$ 40^{\circ} $$ angle), then the triangles are similar.

4. So, a sufficient additional piece of information would be either:
   - There is another pair of corresponding angles that are equal, or
   - The lengths of the two sides forming the $$ 40^{\circ} $$ angle in one triangle and their corresponding sides in the other triangle are in the same ratio.

Any one of these options is sufficient to conclude that the triangles are similar.
A sufficient additional piece of information is knowing that another pair of corresponding angles are equal or that the sides forming the $$40^{\circ}$$ angle are in proportion.

Triangles ABC and DEF have a corresponding angle measuring \( 40^{\circ} \). What other piece of information is sufficient to determine these triangles are similar?

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