Answer
$a=100$ $;$ $b=80$ $;$ $c\approx25.893$
$\angle A=135^{\circ}$ $;$ $\angle B\approx34.45^{\circ}$ $;$ $\angle C\approx10.55^{\circ}$
Work Step by Step
$a=100$ $,$ $b=80$ $,$ $\angle A=135^{\circ}$
The Law of Sines is $\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$
Find angle $B$ by using the formula $\dfrac{\sin A}{a}=\dfrac{\sin B}{b}$, obtained from the Law of Sines. Substitute the known values and solve for $B$:
$\dfrac{\sin135^{\circ}}{100}=\dfrac{\sin B}{80}$
$\sin B=\Big(\dfrac{80}{100}\Big)\sin135^{\circ}$
$\sin B=\Big(\dfrac{4}{5}\Big)\sin135^{\circ}$
$B=\sin^{-1}\Big[\Big(\dfrac{4}{5}\Big)\sin135^{\circ}\Big]\approx34.45^{\circ}$
Two angles are now known. Since $\angle A+\angle B+\angle C=180^{\circ}$, substitute the known angles into the formula and solve for $\angle C$:
$C=180^{\circ}-135^{\circ}-34.45^{\circ}\approx10.55^{\circ}$
Find the side $c$ using the formula $\dfrac{\sin A}{a}=\dfrac{\sin C}{c}$, obtained from the Law of Sines. Substitute the known values and solve for $c$:
$\dfrac{\sin135^{\circ}}{100}=\dfrac{\sin10.55^{\circ}}{c}$
$\dfrac{100}{\sin135^{\circ}}=\dfrac{c}{\sin10.55^{\circ}}$
$c=\Big(\dfrac{\sin10.55^{\circ}}{\sin135^{\circ}}\Big)100\approx25.893$
