53. The three sides of one triangle are $a, a, b$, and the three sides of another triangle are $a, b, b$, where $a>b$. If the smallest interior angles of the two triangles are equal, then the value of $\frac{a}{b}$ is ( ).
A. $\frac{\sqrt{3}+1}{2}$
B. $\frac{\sqrt{5}+1}{2}$
C. $\frac{\sqrt{3}+2}{2}$
D. $\frac{\sqrt{5}+2}{2}$
Answer: B