Let us solve the questions based on the triangle in the image, particularly examining the proportionality of the sides and whether they follow the rules of similar triangles.

We use the Basic Proportionality Theorem (Thales' Theorem), which states that if a line is drawn parallel to one side of a triangle (like $\overline{ED}$), it divides the other two sides proportionally.

Proportions to Verify:

Let’s analyze each statement based on the given triangle:

1. $\frac{AD}{DB} = \frac{AE}{EC}$

This is true by the Basic Proportionality Theorem. The line $\overline{ED}$ is parallel to $\overline{BC}$, so it divides $\overline{AC}$ and $\overline{AB}$ proportionally.

2. $\frac{DB}{AB} = \frac{AE}{AC}$

This is true. Simplify it: $\frac{DB}{AB} = \frac{AE}{AC}.$ Here, $\overline{AB} = \overline{AD} + \overline{DB}$, and the proportions still hold by the Basic Proportionality Theorem.

3. $\frac{AD}{AE} = \frac{DB}{EC}$

This is false. For the Basic Proportionality Theorem, the correct relationships involve the entire lengths of sides, not just parts. Specifically: $\frac{AD}{AE} \neq \frac{DB}{EC}.$

4. $\frac{DB}{AE} = \frac{AD}{EC}$

This is false. The proportions must involve corresponding segments divided by the parallel line $\overline{ED}$, which this does not follow.

5. $\frac{AD}{AB} = \frac{AE}{AC}$

This is true. Both ratios represent corresponding proportional parts of the triangle divided by the parallel line $\overline{ED}$.

6. $\frac{AD}{AB} = \frac{EC}{AC}$

This is false. $\frac{AD}{AB}$ corresponds to $\frac{AE}{AC}$, not $\frac{EC}{AC}$. $EC$ lies on the opposite segment.

Final Answers:

1. True
2. True
3. False
4. False
5. True
6. False

Would you like a detailed explanation of any specific step or concept?

Related Questions:

1. What is the Basic Proportionality Theorem, and how is it applied?
2. How do we prove triangles are similar using proportionality?
3. Why does $\overline{ED} \parallel \overline{BC}$ guarantee proportionality in the triangle?
4. Can you find another set of proportions in this triangle setup?
5. How do parallel lines in a triangle help in proving congruency?

Tip: Always label the known and unknown sides of a triangle carefully when working with proportionality problems!