1956 AHSME Problem 26

Which one of the following combinations of given parts does not determine the indicated triangle?

\begin{array}{l} \textbf{(A)}\ \text{base angle and vertex angle; isosceles triangle} \\ \textbf{(B)}\ \text{vertex angle and the base; isosceles triangle} \\ \textbf{(C)}\ \text{the radius of the circumscribed circle; equilateral triangle} \\ \textbf{(D)}\ \text{one arm and the radius of the inscribed circle; right triangle} \\ \textbf{(E)}\ \text{two angles and a side opposite one of them; scalene triangle} \end{array}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.