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Problem Description Given regular pentagon \(ABCDE\), a circle can be drawn that is tangent to \(\overline{DC}\) at \(D\) and to \(\overline{AB}\) at \(A\). The number of degrees in minor arc \(AD\) is $\textbf{(A)}\ 72 \qquad \textbf{(B)}\ 108 \qquad \textbf{(C)}\ 120 \qquad \textbf{(D)}\ 135 \qquad \textbf{(E)}\ 144$ [asy] size(100); defaultpen(linewidth(0.7)); draw(rotate(18)*polygon(5)); real x=0.6180339887; draw(Circle((-x,0), 1)); int i; for(i=0; i<5; i=i+1) { dot(origin+1*dir(36+72*i)); } label("\(B\)", origin+1*dir(36+72*0), dir(origin--origin+1*dir(36+72*0))); label("\(A\)", origin+1*dir(36+72*1), dir(origin--origin+1*dir(36+72))); label("\(E\)", origin+1*dir(36+72*2), dir(origin--origin+1*dir(36+144))); label("\(D\)", origin+1*dir(36+72*3), dir(origin--origin+1*dir(36+72*3))); label("\(C\)", origin+1*dir(36+72*4), dir(origin--origin+1*dir(36+72*4)));[/asy]

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Solution E

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