In the following figure, the angles marked in gray have the same measure. Similarly, the angles marked in white also have the same measure. Determine the measure of angle $b$.

![](https://cdn.mathpix.com/cropped/2024_05_01_27f67f4520ebb8354fb2g-11.jpg?height=434&width=831&top_left_y=548&top_left_x=710)

#
Solution

Let $x$ and $y$ be the measures of the white and gray angles, respectively, as shown in the figure:

![](https://cdn.mathpix.com/cropped/2024_05_01_27f67f4520ebb8354fb2g-38.jpg?height=502&width=896&top_left_y=457&top_left_x=480)

Remember that the sum of the interior angles of a triangle is always equal to $180^{\circ}$. Adding the three angles of triangle $B C D$, we get

$$
y+b+\left(180^{\circ}-2 x\right)+x=180^{\circ}
$$

Rearranging the equation above, we have

$$
y+b=x
$$

Adding the angles of triangle $A B C$, we get

$$
60^{\circ}+2 y+180^{\circ}-2 x=180^{\circ} .
$$

Rearranging the equation above, we get

$$
x=y+30^{\circ} \text {. }
$$

Thus, we have the system of equations

$$
\left\{\begin{array}{l}
y+b=x \\
y+30^{\circ}=x
\end{array}\right.
$$

from which we conclude that $b=30^{\circ}$.