To solve a triangle means to find the lengths of all its sides and all its interior angles. This is commonly done starting from some pieces of information about the triangle already given, and using the following results:
1.- The sum of the internal angles of every triangle is 180º.
2.- The Law of Sines.
If A and B are the angles opposite to sides with lengths a and b respectively, then:
[tex]\frac{a}{\sin(A)}=\frac{b}{\sin (B)}[/tex]3.- The Law of Cosines
If the sides of a triangle have lengths a, b, c and the angle opposite to the side with length c has a measure C, then:
[tex]c^2=a^2+b^2-2ab\cdot\cos (C)[/tex]In this case, the easiest way to solve the given triangle is to use the fact (1) to find the measure of the angle B. Then, use the Law of Sines to find the measure of the sides BC and AB.
Since B+C+A=180, then:
[tex]\begin{gathered} m\angle B+72.4+48.2=180 \\ \Rightarrow m\angle B+120.6=180 \\ \Rightarrow m\angle B=180-120.6 \\ \Rightarrow m\angle B=59.4 \end{gathered}[/tex]Using the Law of Sines to find BA:
[tex]\begin{gathered} \frac{BA}{\sin (\angle C)}=\frac{AC}{\sin (\angle B)} \\ \Rightarrow BA=\frac{\sin (\angle C)\times AC}{\sin (\angle B)} \\ =\frac{\sin (48.2)\times25.6ft}{\sin (59.4)} \\ =22.172\ldots \end{gathered}[/tex]Using the Law of Sines to find BC:
[tex]\begin{gathered} BC=\frac{\sin (\angle A)\times CA}{\sin (\angle B)} \\ =\frac{\sin (72.4)\times25.6}{\sin (59.4)} \\ =28.3496\ldots \end{gathered}[/tex]Therefore, the missing pieces of information to solve the triangle, are:
[tex]\begin{gathered} m\angle B=59.4º \\ BA=22.2ft \\ BC=28.3ft \end{gathered}[/tex]