3. $A B C$ is an isosceles triangle with $\overline{A C}=\overline{B C}$. A point $A^{\prime}$ is chosen on the side $A C$ such that $\overline{A B}=\overline{B A^{\prime}}=\overline{A^{\prime} C}$. What is the measure of the angle $A \widehat{C} B$?
(A) $30^{\circ}$
(B) $36^{\circ}$
(C) $40^{\circ}$
(D) $45^{\circ}$
(E) $60^{\circ}$
Problem 3. The correct answer is (B).

Let $\alpha=A \widehat{C} B$, then $A^{\prime} \widehat{B} C=\alpha$ ( $B A^{\prime} C$ is isosceles) and thus $A \widehat{A^{\prime}} B=2 \alpha$ (external angle of the triangle $B A^{\prime} C$ ), so $B \widehat{A} C=2 \alpha$ ( $A B A^{\prime}$ is isosceles), as well as $A \widehat{B} C=2 \alpha$. Summing the three angles of $A B C$, we conclude that $5 \alpha=180^{\circ}$, that is, $\alpha=36^{\circ}$.

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