Final answer:
To prove that triangles GHI and JKL are similar, we need to either know the measures of the other angles or the lengths of the sides to apply the AA or SSS similarity criteria, respectively.
Step-by-step explanation:
If triangle GHI is dilated to create triangle JKL on a coordinate grid, and it's given that angle H is congruent to angle K, to prove that the two triangles are similar, the most straightforward method is to show that all corresponding angles are congruent and that the sides are proportional. However, with just one angle congruent, we need more information. We need to either know the measures of the other angles in GHI (the measure of angle G or the measure of angle I), thereby proving that all corresponding angles are congruent, or we need to know the sides of GHI or JKL to establish the proportional relationship between the corresponding sides of both triangles.
According to the properties of similar triangles, if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. This is known as the AA (Angle-Angle) similarity postulate. Thus, by knowing the measure of another angle in either triangle, we can apply this postulate. Alternatively, if we know the corresponding sides, we can use the Side-Side-Side (SSS) similarity theorem if all three pairs of corresponding sides are in proportion.