Similar Triangles ADE and ABC: Analyzing Geometric Relationships

To solve this problem, we will use the properties of similar triangles. In similar triangles, the corresponding sides are proportional. This means if triangles ADE and ABC are similar, the ratios of these corresponding sides must be equal. We can set up the proportion by considering:

• Side $AD$ in triangle ADE corresponds to side $AB$ in triangle ABC.
• Side $AE$ in triangle ADE corresponds to side $AC$ in triangle ABC.
• Side $DE$ in triangle ADE corresponds to side $BC$ in triangle ABC.

Therefore, based on this correspondence, the ratio of the sides should be:

$\frac{AD}{AB}=\frac{AE}{AC}=\frac{DE}{BC}$

To confirm, let's ensure we are interpreting the similarity correctly:

• Triangle ADE is similar to triangle ABC, meaning their corresponding angles are equal, and sides track on a consistent ratio.

Thus, based on the properties of similar triangles, the correct answer corresponding to these proportion relationships is:

$\frac{AD}{AB}=\frac{AE}{AC}=\frac{DE}{BC}$

This corresponds to choice 4.