[ Equilateral (regular) triangle ] [ Rotations by $\$ 60$ ^ \circ $\$$ and $\$ 120$ ^\circ\$

In rhombus $A B C D$, angle $A B C$ is $120^{\circ}$. Points $P$ and $Q$ are taken on sides $A B$ and $B C$, respectively, such that $A P=B Q$. Find the angles of triangle $P Q D$.

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Consider a rotation by $60^{\circ}$ around point $D$.

## Solution

Triangles $A B D$ and $C B D$ are equilateral. When rotating around point $D$ by an angle of $60^{\circ}$, which maps vertex $A$ to vertex $B$, vertex $B$ transitions to vertex $C$, side $A B$ to side $B C$. Since $A P = B Q$, point $P$ transitions to point $Q$ during this rotation. Therefore, triangle $P Q D$ is equilateral.

## Answer

$60^{\circ}, 60^{\circ}, 60^{\circ}$.