The angle bisectors of angles $\beta$ and $\gamma$ in triangle $ABC$ form an angle $\varphi$. The ratio of the angles is given as $\alpha : \varphi = 1 : 2$ and $\beta : \gamma = 1 : 4$. We are asked to find the interior angles of the triangle.

1. Problem Description

The angle bisectors of angles $\beta$ and $\gamma$ in triangle $ABC$ form an angle $\varphi$. The ratio of the angles is given as $\alpha : \varphi = 1 : 2$ and $\beta : \gamma = 1 : 4$. We are asked to find the interior angles of the triangle.

2. Solution Steps

Let the angles of the triangle be $\alpha$, $\beta$, and $\gamma$. We are given that $\beta : \gamma = 1 : 4$, so $\gamma = 4\beta$. The sum of the angles in a triangle is $180^{\circ}$, so

$\alpha + \beta + \gamma = 180^{\circ}$.

Substituting $\gamma = 4\beta$, we have

$\alpha + \beta + 4\beta = 180^{\circ}$

$\alpha + 5\beta = 180^{\circ}$.

Let the angle bisectors of $\beta$ and $\gamma$ intersect at a point. Let $\varphi$ be the angle between these angle bisectors.

The angle between the angle bisectors is given by

$\varphi = 180^{\circ} - \frac{\beta}{2} - \frac{\gamma}{2} = 180^{\circ} - \frac{\beta}{2} - \frac{4\beta}{2} = 180^{\circ} - \frac{5\beta}{2}$.

We are given that $\alpha : \varphi = 1 : 2$, so $\varphi = 2\alpha$. Substituting this into the previous equation, we get

$2\alpha = 180^{\circ} - \frac{5\beta}{2}$.

$4\alpha = 360^{\circ} - 5\beta$.

$4\alpha + 5\beta = 360^{\circ}$.

Now we have a system of two equations with two variables:

$\alpha + 5\beta = 180^{\circ}$

$4\alpha + 5\beta = 360^{\circ}$

Subtracting the first equation from the second equation, we get

$3\alpha = 180^{\circ}$, so $\alpha = 60^{\circ}$.

Substituting $\alpha = 60^{\circ}$ into the first equation, we get

$60^{\circ} + 5\beta = 180^{\circ}$

$5\beta = 120^{\circ}$

$\beta = 24^{\circ}$.

Since $\gamma = 4\beta$, we have $\gamma = 4(24^{\circ}) = 96^{\circ}$.

So the angles are $\alpha = 60^{\circ}$, $\beta = 24^{\circ}$, and $\gamma = 96^{\circ}$.

3. Final Answer

$\alpha = 60^{\circ}$, $\beta = 24^{\circ}$, $\gamma = 96^{\circ}$.