정사각형과 작도된 선분의 교차점에서 생기는 각의 크기

문제

그림과 같이 정사각형 $ABCD$의 변 $AB$의 중점을 $M$, 변 $BC$의 중점을 $N$이라고 하자. 두 선분 $DM$과 $AN$의 교점을 $P$라고 할 때, $\angle APD$의 크기는 몇 도인가?

(단, 그림은 이해를 돕기 위한 것이며, 실제 각도나 길이에 대한 정확한 비례는 보장하지 않는다.)

graph TD
    A[A] --- B(B)
    B --- C(C)
    C --- D(D)
    D --- A
    A --- M(M)
    M --- B
    B --- N(N)
    N --- C
    D --- P(P)
    A --- P
    P --- N
    P --- M

    style A fill:#fff,stroke:#333,stroke-width:2px,rx:5px,ry:5px;
    style B fill:#fff,stroke:#333,stroke-width:2px,rx:5px,ry:5px;
    style C fill:#fff,stroke:#333,stroke-width:2px,rx:5px,ry:5px;
    style D fill:#fff,stroke:#333,stroke-width:2px,rx:5px,ry:5px;
    style M fill:#fff,stroke:#333,stroke-width:2px,rx:5px,ry:5px;
    style N fill:#fff,stroke:#333,stroke-width:2px,rx:5px,ry:5px;
    style P fill:#fff,stroke:#333,stroke-width:2px,rx:5px,ry:5px;

    linkStyle 0 stroke-width:2px,fill:none,stroke:black;
    linkStyle 1 stroke-width:2px,fill:none,stroke:black;
    linkStyle 2 stroke-width:2px,fill:none,stroke:black;
    linkStyle 3 stroke-width:2px,fill:none,stroke:black;
    linkStyle 4 stroke-width:2px,fill:none,stroke:black,stroke-dasharray: 5 5;
    linkStyle 5 stroke-width:2px,fill:none,stroke:black,stroke-dasharray: 5 5;
    linkStyle 6 stroke-width:2px,fill:none,stroke:black,stroke-dasharray: 5 5;
    linkStyle 7 stroke-width:2px,fill:none,stroke:black;
    linkStyle 8 stroke-width:2px,fill:none,stroke:black;
    linkStyle 9 stroke-width:2px,fill:none,stroke:black;