Is this possible to solve?

Question

Is this possible to solve the length of $AB$ while only knowing that $l_1$ and $l_2$ are two parallel lines?

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Example with geogebra

Answer 1

This question is tricky!

While it is tempting to use the Intercept Theorem, and calculate $AB$ from the following equation:

$$\frac{DB}{DE}=\frac{AB}{FE}$$

This would yield a wrong answer. Raffaele's answer explains why very clearly.

Answer 2

Sine law in triangle $ABD$

$\gamma=\beta=37°$ because $l_1\parallel l_2$ $$\frac{AB}{\sin 55^{\circ}}=\frac{25}{\sin (180^{\circ}-55^{\circ}-37^{\circ})}\to AB\approx 20.4913$$

edit

The segment $FE$ in the original drawing is a deception. As you can see from the calculation above, $FE$ is not necessary to find $AB$. Furthermore $FE=39$ would make points $D,A,F$ not aligned because $F'E$ is larger than $FE$.

Hope this is clear

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