Angles in a triangle

In a triangle, the first angle measures a number plus 45°, and the second angle is 30° less than the first angle. The third angle is two times the number more than the first angle. What are the measurements of each angle of the triangle?

Final Answer:

α =  63.75 °
β =  33.75 °
γ =  82.5 °

Step-by-step explanation:

$$\begin{gathered} \alpha =2x+45 \\ \beta =\alpha -30=2x+45-30=2x+15 \\ \gamma =2x+\alpha =2x+2x+45=4x+45 \\ \alpha +\beta +\gamma =180 \\ 2x+45+2x+15+(4x+45)=180 \\ 2\cdot x+45+2\cdot x+15+(4\cdot x+45)=180 \\ 8x=75 \\ x=\frac{75}{8}=9.375 \\ x=\frac{75}{8}=9.375 \\ \alpha =2\cdot x+45=2\cdot 9.375+45=63.75^{\circ }=63°45^{\prime } \end{gathered}$$

$\beta =2\cdot x+15=2\cdot 9.375+15=33.75^{\circ }=33°45^{\prime }$

$$\begin{gathered} \gamma =4\cdot x+45=4\cdot 9.375+45=\frac{165}{2}=82.5=82.5^{\circ }=82°30^{\prime } \\ \text{ Verifying Solution: } \\ {s}_1=\alpha +\beta +\gamma =63.75+33.75+82.5=180{}^{\circ } \end{gathered}$$

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