$\Delta ABC\sim\Delta EDC$: $\overline{AB}$ is parallel to $\overline{DE}$ is true. How is $\overline{AE}$ is not perpendicular to $\overline{BD}$?

Question

In the figure below, $\Delta ABC\sim\Delta EDC$. Which of the following must be true?

1. $\overline{AE}$ is parallel to $\overline{BD}$
2. $\overline{AE}$ is perpendicular to $\overline{BD}$
3. $\overline{AB}$ is parallel to $\overline{DE}$
4. $\overline{AB}$ is perpendicular to $\overline{DE}$

I can see that $\overline{AB}$ is parallel to $\overline{DE}$. However, $\overline{AE}$ perpendicular to $\overline{BD}$ seems feasible too. However, the correct answer is said to be only option 3. The reasoning for option 2 being wrong has been mentioned as:

The answer may result from visual inspection of the diagram. The line segments appear to be perpendicular, but need not be, given the information provided.

How are these lines not perpendicular? What am I missing?

Answer 1

The question says which of the following must be true.

Note that if $AE$ is perpendicular to $BD$, the conditions given in the question are followed. However, $AE$ may not be perpendicular to $BD$ and $\Delta ABC$ might still be similar to $\Delta EDC$.

The question conditions neither claim nor deny the occurence of perpendicularity; $90°$ is just one of the many possibilities (for $\angle ACB$) and not a must-happen event. $\angle ACB<90°$ is also possible. If it is still not clear to you try putting $\angle ABC=40°$ and $\angle CAB=80°$. What happens when $\Delta ABC\sim\Delta EDC$?