Three squares are attached to each other by their vertices and to two vertical rods, as shown in the figure. Determine the measure of angle $x$.

![](https://cdn.mathpix.com/cropped/2024_05_01_96f63abf6bdef97495eag-34.jpg?height=388&width=656&top_left_y=1847&top_left_x=571)

#
Solution

In the drawing below, where $AB$ is parallel to $CD$, we will show that the sum of the white angles is equal to the sum of the measures of the gray angles. This result holds for any number of "peaks" in the drawing and is popularly known as the "Theorem of Peaks".

![](https://cdn.mathpix.com/cropped/2024_05_01_96f63abf6bdef97495eag-35.jpg?height=381&width=278&top_left_y=605&top_left_x=966)

At each vertex of the "peaks," draw a line parallel to the segment $AB$. Notice that several pairs of alternate interior angles, which are congruent, will be formed as indicated in the figure below.

![](https://cdn.mathpix.com/cropped/2024_05_01_96f63abf6bdef97495eag-35.jpg?height=495&width=343&top_left_y=1307&top_left_x=931)

Each of the marked angles has exactly one representative among the white and gray angles. Thus, each of these sums of angle measures is $x_{1}+x_{2}+\ldots+x_{6}$.

Since the two vertical bars are parallel, we can apply the Theorem of Peaks to the polygonal path formed by the sides of the squares containing the marked angles, obtaining:

$$
30^{\circ}+126^{\circ}+75^{\circ}+x=90^{\circ}+90^{\circ}+90^{\circ}
$$

Thus, $x=39^{\circ}$.