Question

the second angle of a triangle is 4 times as large as the first. the third angle is 80 degrees more then the sum of the other two angels. find the measure of all three angels. please show allwork

Answer 1

From your problem we have,
\[\alpha _{2}=4\times \alpha _{1}\]\[\alpha _{3}=\alpha _{2}+\alpha _{1}+80\]Because it is a triangle we have also\[\alpha _{1}+\alpha _{2}+\alpha _{3}=180\]Reordering,\[-4\times \alpha _{1}+\alpha_{2}=0\]\[\alpha_{1}+\alpha_{2}-\alpha_{3}=-80\]\[\alpha _{1}+\alpha _{2}+\alpha _{3}=180\]Converting to matrix,\[\left[\begin{matrix}-4 & 1 & 0 &0\\ 1 & 1 & -1&-80 \\ 1 & 1 & 1 & 180\end{matrix}\right]=\left[\begin{matrix}1 & 0 & 0 & 10 \\ 0 & 1 & 0 & 40 \\ 0 & 0 & 1 & 130\end{matrix}\right]\]Thus,\[\alpha _{1}=10 , \alpha _{2}=40 , \alpha _{3}=130\]
Cheers.