1954 AHSME Problem 50

The times between 7 and 8 o'clock, correct to the nearest minute, when the hands of a clock will form an angle of 84^{\circ} are:

\begin{array}{l} \textbf{(A)}\ \text{7: 23 and 7: 53}\qquad \textbf{(B)}\ \text{7: 20 and 7: 50}\qquad \textbf{(C)}\ \text{7: 22 and 7: 53}\\ \textbf{(D)}\ \text{7: 23 and 7: 52}\qquad \textbf{(E)}\ \text{7: 21 and 7: 49} \end{array}

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