# if anglec=angleb and anglea=4angleb, find the value of each angle.

Non-right triangles and trigonometry
if $$\displaystyle\angle{c}=\angle{b}{\quad\text{and}\quad}\angle{a}={4}\angle{b}$$, find the value of each angle.

2020-12-07
The sum of all angles in any triangles is $$\displaystyle{180}^{\circ}$$, so
$$\displaystyle\angle{a}+\angle{b}+\angle{c}={180}$$
Substitute $$\displaystyle\angle{c}=\angle{b}{\quad\text{and}\quad}\angle{a}={4}\angle{b}$$ to solve for $$\displaystyle\angle{b}$$:
$$\displaystyle{4}\angle{b}+\angle{b}+\angle{b}={180}^{\circ}$$
$$\displaystyle{6}\angle{b}={180}^{\circ}$$
$$\displaystyle\angle{b}=\frac{{{180}^{\circ}}}{{6}}$$
$$\displaystyle\angle{b}={30}^{\circ}$$
Substitute $$\displaystyle\angle{c}=\angle{b}$$ to solve for $$\displaystyle\angle{c}$$:
$$\displaystyle\angle{c}={30}^{\circ}$$
Substitute $$\displaystyle\angle{a}={4}\angle{b}\to$$ solve for $$\displaystyle\angle{a}$$:
$$\displaystyle\angle{a}={4}\times{30}^{\circ}$$
$$\displaystyle\angle{a}={120}^{\circ}$$
$$\displaystyle\angle{b}={30}^{\circ},\angle{c}={30}^{\circ},\angle{a}={120}^{\circ},$$