Question

if $gamma=\beta$ and $alpha = 4\beta$, find all three angles in the following triangle. $alpha+\beta+gamma = 180^{circ}$ $alpha=$ $\beta=$ $gamma=$

Explanation:

Step1: Substitute angle - relationships into sum - of - angles formula

Given $\gamma=\beta$ and $\alpha = 4\beta$, and the sum of angles in a triangle $\alpha+\beta+\gamma = 180^{\circ}$. Substitute $\alpha$ and $\gamma$: $4\beta+\beta+\beta=180^{\circ}$.

Step2: Simplify the left - hand side of the equation

Combine like terms: $6\beta = 180^{\circ}$.

Step3: Solve for $\beta$

Divide both sides by 6: $\beta=\frac{180^{\circ}}{6}=30^{\circ}$.

Step4: Find $\alpha$

Since $\alpha = 4\beta$, then $\alpha=4\times30^{\circ}=120^{\circ}$.

Step5: Find $\gamma$

Since $\gamma=\beta$, then $\gamma = 30^{\circ}$.

Answer:

$\alpha = 120^{\circ}$
$\beta = 30^{\circ}$
$\gamma = 30^{\circ}$