# Chapter 7 - Test - Page 353: 6a

If $b \gt 10$, then $0 \lt A \lt 30^{\circ}$ and the angle $A$ has exactly one value.

#### Work Step by Step

Since $B = 150^{\circ}$, then $0 \lt A \lt 30^{\circ}$, otherwise, no such triangle exists. We can use the law of sines to determine the values for $b$: $\frac{b}{sin~B} = \frac{a}{sin~A}$ $sin~A = \frac{a~sin~B}{b}$ $sin~A = \frac{10~sin~150^{\circ}}{b}$ $sin~A = \frac{5}{b}$ Since $sin~30^{\circ} = 0.5$, then $sin~A \lt 0.5$ $\frac{5}{b} \lt 0.5$ $b \gt 10$ If $b \gt 10$, then $0 \lt A \lt 30^{\circ}$ and the angle $A$ has exactly one value.

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