[ Rectangles and Squares. Properties and Characteristics]

Inside the square $A B C D$, a point $M$ is taken such that $\angle M A B=60^{\circ}, \angle M C D=15^{\circ}$. Find $\angle M B C$.
Construct an equilateral triangle on $A B$ inside the square.

## Solution

Construct an equilateral triangle $A B K$ on $A B$ inside the square. Then the point $M$ lies on the ray $A K$. Moreover,

$$
\begin{gathered}
\angle K B A=60^{\circ}, \angle K B C=30^{\circ}, B C=B K, \\
\angle B C K=\frac{1}{2}\left(180^{\circ}-30^{\circ}\right)=75^{\circ}, \angle K C D=15^{\circ} .
\end{gathered}
$$

Therefore, the point $M$ lies on the ray $C K$. Consequently, the point $K$ coincides with the point $M$.

![](https://cdn.mathpix.com/cropped/2024_05_06_9466e5d28b3947d5c4a5g-12.jpg?height=414&width=506&top_left_y=1548&top_left_x=776)

Answer

$30^{\circ}$.

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