We divided a rectangle into four triangles as shown in the picture. We measured all the internal angles in these triangles and obtained the following values: $15^{\circ}, 20^{\circ}, 20^{\circ}, 50^{\circ}, 55^{\circ}, 70^{\circ}, 75^{\circ}, 75^{\circ}, 90^{\circ}, 90^{\circ}, 130^{\circ}$, and one value that we forgot to write down. Determine the missing value and specify which angle it is, given that segment $B E$ is longer than segment $F C$. (Note, the picture may be inaccurate, it is not worth measuring.)

(S. Bednářová)

![](https://cdn.mathpix.com/cropped/2024_04_17_fa77236f1bc7a0516474g-1.jpg?height=462&width=1131&top_left_y=1962&top_left_x=494)
First, we will fill in the angles that are "obvious."

![](https://cdn.mathpix.com/cropped/2024_04_17_fa77236f1bc7a0516474g-2.jpg?height=457&width=1128&top_left_y=234&top_left_x=498)

The sum of the remaining angles in triangle $E C F$ is $50^{\circ}$. From the angles available, 50 cannot be obtained, so the missing angle must be in this triangle. The sum of the remaining angles in triangle $D E F$ is $130^{\circ}$, which can be obtained as the sum of 75 and 55. Since $|\nless D E A| = |\Varangle E D F|$, it must be that $|\Varangle D E A| = 75^{\circ}$. We will fill in this information on the diagram, calculate some angles, and get:

![](https://cdn.mathpix.com/cropped/2024_04_17_fa77236f1bc7a0516474g-2.jpg?height=459&width=1125&top_left_y=1027&top_left_x=497)

There are still angles $20^{\circ}, 20^{\circ}, 70^{\circ}, x^{\circ}$ to be filled in. Since $|\nless E C F| = |\nless C E B|$, it must be that $|\Varangle C E B| = 20^{\circ}$. Now we can easily calculate that the missing angle is $30^{\circ}$, and fill in the diagram.

![](https://cdn.mathpix.com/cropped/2024_04_17_fa77236f1bc7a0516474g-2.jpg?height=454&width=1128&top_left_y=1680&top_left_x=498)

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