Given that the triangles in Figure 19 are similar, find the missing length $x$ and angles $A$, $B$, and $C$.

Figure 19

Solution Because the triangles are similar, corresponding angles have the same measure. So, $A=71\mathbf{ {{}^\circ}}$, $B=19\mathbf{ {{}^\circ}}$, and $C=90\mathbf{ {{}^\circ}}$. Also corresponding sides are proportional. That is, $\dfrac{3}{5}=\dfrac{ 9}{x}.$ We solve this equation for $x$. $$\begin{array}{r@{ }c@{ }l@{}l@{}l} \dfrac{3}{5} &=& \dfrac{9}{x} & & \\ 5x\cdot \dfrac{3}{5} &=& 5x\cdot \dfrac{9}{x} & & {\color{#0066A7}{\hbox{Multiply both sides by \(5x\).}}} \\ 3x&=&45 & & {\color{#0066A7}{\hbox{Simplify.}}} \\[0pt] x&=&15 & & {\color{#0066A7}{\hbox{Divide both sides by 3.}}} \end{array}$$ The missing length is $15$ units.