Three angles

In a triangle ABC, the magnitude of the internal angle gamma is equal to one-third of the angle alpha. The size of the angle beta is 80 degrees larger than the size of the gamma angle. Calculate the magnitudes of the interior angles of the triangle ABC.

Final Answer:

α =  60 °
β =  100 °
γ =  20 °

Step-by-step explanation:

$$\begin{gathered} \alpha +\beta +\gamma =180 \\ \gamma =\alpha /3 \\ \beta =80+\gamma \\ \alpha +(80+\gamma )+\gamma =180 \\ \alpha +80+2\gamma =180 \\ \alpha +80+2\cdot (\alpha /3)=180 \\ 5\alpha =300 \\ \alpha =\frac{300}{5}=60 \\ \alpha =60=60^{\circ } \end{gathered}$$

$$\begin{gathered} \gamma =\alpha /3=60/3=20{}^{\circ } \\ \beta =80+\gamma =80+20=100=100^{\circ } \\ \text{ Verifying Solution: } \\ s=\alpha +\beta +\gamma =60+100+20=180{}^{\circ } \end{gathered}$$

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