Exercise 30 Page 517

We are given that in the figure $\angle A\cong \angle C$ and asked which additional information would not be enough to prove that $△ADB$ and $△CEB$ are similar.

Let's consider each of the given options separately.

Option F

In this option, we also know that the ratio of corresponding sides is equal for both triangles. We can assume that $\angle ABE$ and $\angle CBE$ are right angles. If the ratio of at least two corresponding sides is equal for two triangles and the included angle is congruent, then these triangles are similar. Therefore, this answer is not correct.

Option G

Now, our additional information is that $\angle ADB\cong \angle CEB.$ This is enough information to state that these triangles are similar by the Angle-Angle Similarity Theorem. Therefore, this answer is not correct.

Option H

Here we are given that $\overline{ED}$ and $\overline{DB}$ are congruent.

This means that the length of $\overline{EB}$ is two times the length of $\overline{DB}.$ However, as we know nothing about the other sides of these triangles we cannot determine if they are similar. Therefore, answer H is correct.

Option J

Finally, we are given that $\overline{BC}$ is perpendicular to $\overline{AC}.$ This means that $\angle ABD$ and $\angle CBE$ are right angles.

As we can see, $△ADB$ and $△CEB$ have two angles that are congruent. Therefore, by the Angle-Angle Similarity Theorem, these angles are similar and answer J is not correct.