Al-T?si demonstrated a method to solve a spherical triangle if all three angles are known. Suppose the three angles of triangle $A B C$ are given (Fig. 9.39), where we assume that all three sides of the triangle are less than a quadrant. We extend each side of the triangle two different ways to form a quadrant. That is, we extend $A B$ to $A D$ and $B H$; $A C$. to $A E$ and $C G$; and $B C$ to $B K$ and $F C$, where all of the six new arcs are quadrants. We then draw great circle arcs through $D$ and $E, F$ and $G$, and $H$ and $K$ to form the new spherical triangle $L M N$. Now the vertices of the original triangle are the poles of the three sides of the new triangle. Then, for example, $M D=E N=90^{\circ}-D E=90^{\circ}-A$, or $M N=180^{\circ}-A$. Thus, the three sides of triangle $L M N$ are known, and therefore the triangle can be solved by the procedure sketched in the text. But we also know that the vertices of triangle $L M N$ are the poles of the original triangle. So, for example, $B F=C K=90^{\circ}-B C$, and $L=F K=180^{\circ}-B C$. We therefore can determine the sides of the original triangle. Use this procedure to solve the triangle $A B C$, where $A=75^{\circ}, B=80^{\circ}$, and $C=85^{\circ}$.