1961 AMC 12 第 25 题

Let $\angle B=m$ and use each pair of equal segments to create isosceles triangles

Chase the angles through triangles $BQP,$ $AQP,$ and $APC$

Let $\angle B=m.$ Since $PQ=QB,$ triangle $BQP$ gives $\angle QPB=m$ and $\angle AQP=2m.$ Since $AP=PQ,$ triangle $AQP$ gives $\angle QAP=2m,$ so $\angle QPA=180^\circ-4m.$ As $B,P,C$ are collinear, $\angle APC=3m.$ Now $AP=AC$ gives $\angle ACP=3m.$ Finally $AB=BC$ makes both base angles of $ABC$ equal to $3m,$ and $$m+3m+3m=180^\circ.$$ Therefore $m=\frac{180^\circ}{7}=25\dfrac57^\circ.$

Thus, the correct answer is A.