In triangle $ABC$, point $D$ lies on side $AC$ and point $E$ lies on side $BC$. The measures of angles $ABD, BAE, CAE$, and $CBD$ are $30^{\circ}, 60^{\circ}, 20^{\circ}$, and $30^{\circ}$, respectively.

Determine the measure of angle $AED$.

![](https://cdn.mathpix.com/cropped/2024_04_17_6d7ca75cbeb5035abbabg-4.jpg?height=525&width=528&top_left_y=443&top_left_x=770)

Note: The image is for illustration only.

(A. Bohiniková)

Hint. Look for congruent angles.
The size of angle $ABC$ is equal to the sum of the sizes of angles $ABD$ and $CBD$, i.e., $30^{\circ}+30^{\circ}=60^{\circ}$. In triangle $ABE$, two interior angles have a size of $60^{\circ}$, so the remaining angle also has a size of $60^{\circ}$, making the triangle equilateral.

Line $BD$ is the axis of the interior angle of the equilateral triangle $ABE$, and thus it is also its height. In particular, points $A$ and $E$ are symmetric with respect to line $BD$. Segments $AD$ and $ED$, as well as angles $EAD$ and $AED$, are therefore congruent. The size of angle $AED$ is $20^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_04_17_6d7ca75cbeb5035abbabg-4.jpg?height=668&width=488&top_left_y=1511&top_left_x=790)

Note. According to the data given in the problem, triangle $ABC$ can be constructed, including points $D$ and $E$. This is a good starting point for uncovering the necessary relationships. However, a solution based on measurements from the image cannot be considered satisfactory.

On this occasion, we would like to remind you that an angle of $20^{\circ}$ (like many other angles) cannot be constructed using only a ruler and a compass.