An exterior angle of a triangle is $108^{\circ}$ and its interior opposite angles are in the ratio 4: 5. The angles of the triangle are:

Question

Accepted Answer

To determine the angles of the triangle, we start by understanding the relationship between the exterior angle and the interior opposite angles. The exterior angle of a triangle is equal to the sum of the two interior opposite angles. In this problem, the exterior angle is given as $108^\circ$.

The interior opposite angles are in the ratio 4:5. Let's denote these angles as $4x$ and $5x$. According to the exterior angle theorem:

$$ 4x + 5x = 108^\circ $$

This simplifies to:

$$ 9x = 108^\circ $$

Solving for $x$, we get:

$$ x = \frac{108^\circ}{9} = 12^\circ $$

Now, we can find the measures of the two interior opposite angles:

- The first angle is $4x = 4 \times 12^\circ = 48^\circ$
- The second angle is $5x = 5 \times 12^\circ = 60^\circ$

The third angle of the triangle can be found using the fact that the sum of all interior angles in a triangle is $180^\circ$:

$$ 48^\circ + 60^\circ + \text{third angle} = 180^\circ $$

$$ \text{third angle} = 180^\circ - 108^\circ = 72^\circ $$

Thus, the angles of the triangle are $48^\circ$, $60^\circ$, and $72^\circ$, which corresponds to Option 1: $48^\circ, 60^\circ, 72^\circ$.

Now, let's explain why the other options are incorrect:

- **Option 2: $50^\circ, 60^\circ, 70^\circ$**: The sum of these angles is $180^\circ$, but they do not satisfy the condition that the exterior angle ($108^\circ$) is the sum of the two interior opposite angles. $50^\circ + 60^\circ = 110^\circ$, which does not match the given exterior angle.

- **Option 3: $52^\circ, 56^\circ, 72^\circ$**: The sum of these angles is $180^\circ$, but the two interior opposite angles $52^\circ$ and $56^\circ$ add up to $108^\circ$, which matches the exterior angle. However, the ratio of these angles is not 4:5.

- **Option 4: $42^\circ, 60^\circ, 76^\circ$**: The sum of these angles is $178^\circ$, which is incorrect for a triangle. Additionally, the sum of the two interior opposite angles $42^\circ$ and $60^\circ$ is $102^\circ$, which does not match the given exterior angle of $108^\circ$.

Therefore, Option 1 is the only choice that satisfies all the conditions given in the problem, making it the correct answer.