7. Square $A B C D$ has a side length of $1, A B, A D$ each have a point $P, Q$, if the perimeter of $\triangle A P Q$ is 2, find the degree of $\angle P C Q$.

The square $A B C D$ has a side length of 1, and points $P$ and $Q$ are on $A B$ and $A D$ respectively. If the perimeter of $\triangle A P Q$ is 2, find the degree of $\angle P C Q$.
7. Rotate $\triangle Q D C$ around point $C$ by $90^{\circ}$ (counterclockwise) to $\triangle Q^{\prime} B C$. From the given information, we know that $A P + A Q + P Q = 2 = A B + A D = A P + P B + A Q + Q D$, so $P Q = P B + Q D = P B + B Q^{\prime} = P Q^{\prime}$.

Also, $P C = P C, Q C = Q^{\prime} C$, thus $\triangle P Q^{\prime} C \cong \triangle P Q C$. Therefore, $\angle P C Q = \angle P C Q^{\prime}$, and $\angle P C Q + \angle P C Q^{\prime} = \angle B C Q + \angle B C Q^{\prime} = \angle B C Q + \angle Q C D = 90^{\circ}$. From this, we deduce that $\angle P C Q = 45^{\circ}$.