a =72.97 and c=8.62
a) To find out the value of a let's use the Law of Sines. This way:
[tex]\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}[/tex]
And the fact that the sum of the interior angles within a triangle is always 180º so, m∠B =180-(80+35) , m∠B = 65º. Now let's plug that information into that:
[tex]\begin{gathered} \frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)} \\ \frac{a}{\sin(80)}=\frac{42.5}{\sin(35)} \\ a\sin (35)=42.5\cdot\sin (80) \\ a=\frac{42.5\cdot\sin(80)}{\sin(35)} \\ a\approx72.97 \end{gathered}[/tex]
b) Let's find the value of that angle A, at first. m∠A =180-(52+26), m∠A=102º
Likewise, we're going to adopt the same way to find the measure of leg c:
[tex]\begin{gathered} \frac{19.3}{\sin(102)}=\frac{c}{\sin (26)} \\ c=\frac{19.3\cdot\sin (26)}{\sin (102)} \\ c\approx8.65 \end{gathered}[/tex]
2) Hence, the answers are a =72.97 and c=8.62
