The problem asks to find the value of angle $q$ in the given diagram. We are given an exterior angle of $235^{\circ}$ and two interior angles of a triangle ($47^{\circ}$ and $24^{\circ}$).

1. Problem Description

The problem asks to find the value of angle $q$ in the given diagram. We are given an exterior angle of $235^{\circ}$ and two interior angles of a triangle ($47^{\circ}$ and $24^{\circ}$).

2. Solution Steps

First, we need to find the interior angles adjacent to the exterior angle. We know that the exterior angle and the adjacent interior angle form a straight line, which means they add up to $180^{\circ}$. Thus, the interior angle adjacent to the $235^{\circ}$ exterior angle is:

$180^{\circ} - (360^{\circ} - 235^{\circ}) = 180^{\circ} - 125^{\circ} = 55^{\circ}$.

Alternatively, consider the angles adjacent to 47 and 24 respectively. We are given that the angles adjacent to these form 235 degrees. The exterior angle is equal to the sum of the two remote interior angles. Thus $235^\circ$ is the exterior angle formed between these two vertices and their adjacent vertex. The interior angles are 47 and

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4. Let's compute the other two angles adjacent to 47 and 24 respectively.

$180 - 47 = 133$ and $180 - 24 = 156$ so $133+156 = 289$, but that isn't $235$. This means 235 is the exterior angle.

Instead, let's compute the internal angles. Since an exterior angle is $235^{\circ}$, the adjacent interior angle is $360^{\circ}-235^{\circ} = 125^{\circ}$.

So the triangle has angles with two being $47^{\circ}$ and $24^{\circ}$, and one angle adds with $235^\circ$ to be $360^\circ$ which means the interior angle is $360^\circ-235^\circ = 125^\circ$. However, we need the angle formed inside between the sides. This is equivalent to $180^\circ - (360^\circ - 235^\circ) = 180^\circ - 125^\circ = 55^\circ$

Then, the sum of angles in a triangle is $180^{\circ}$. The three interior angles are $47^{\circ}$, $24^{\circ}$, and $55^{\circ}$.

$180^{\circ} - 47^{\circ} - 24^{\circ}= 109^\circ$.

$125^{\circ} = 180^\circ - x$, or $x = 180^\circ-235^\circ$ and that doesn't make sense

$180 - 47 = a$ and $180 - 24 = b$ and $a + b = 235$, then $a = 235 - b = 235 - (180 - 24) = 235 - 156 = 79$ and $b = 235 - a = 235 - (180 - 47) = 235 - 133 = 102$. Thus the angles would be 79 + 102 = 235 which sums to $180 + 180$.

Now, we have a triangle with angles $47^{\circ}$ and $24^{\circ}$. We need to find the third angle which adjacent to $235^{\circ}$. The angle inside is $360^{\circ} - 235^{\circ} = 125^{\circ}$. This is incorrect.

Let $x, y, z$ be the interior angles of the triangle. $x = 47^{\circ}, y = 24^{\circ}$. Then, $x + y + q = 180^{\circ}$. Also, the exterior angle adjacent to angle $q$ is $235$. This is incorrect.

The exterior angle is the sum of the opposite interior angles.

Thus, $235^\circ$. But this isn't a proper exterior angle.

Let $A, B, C$ denote angles in the triangle. We are given $A = 47^\circ$ and $B = 24^\circ$.

Thus $C = 180^\circ - (47^\circ + 24^\circ) = 180^\circ - 71^\circ = 109^\circ$.

The exterior angle to $C$ would be $180^\circ - 109^\circ = 71^\circ$. Which isn't

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5.

Let's find the third angle in the triangle. Since the angles in a triangle add to 180, and we know that $47 + 24 = 71$, the interior angle is $180-71 = 109$. So angle Q is $109^{\circ}$. $360^\circ - 235^\circ$

Thus $47 + 24 = 71$ and $q = 180-71=109$

3. Final Answer

109