We are given a diagram with a triangle and an exterior angle. The triangle has two sides that are marked as equal, so it's an isosceles triangle. We need to find the value of $k$, which represents an angle inside the triangle. An exterior angle of $83^{\circ}$ is also given.

1. Problem Description

We are given a diagram with a triangle and an exterior angle. The triangle has two sides that are marked as equal, so it's an isosceles triangle. We need to find the value of $k$, which represents an angle inside the triangle. An exterior angle of $83^{\circ}$ is also given.

2. Solution Steps

First, let's analyze the isosceles triangle. Since two sides are equal, the angles opposite those sides are also equal. One of these angles is $k$.

The exterior angle and $k$ are supplementary, so their sum is $180^{\circ}$. Thus:

$k + 83^{\circ} = 180^{\circ}$

$k = 180^{\circ} - 83^{\circ}$

$k = 97^{\circ}$

However, if the triangle is isosceles, let the other equal angle also be denoted as $x$.

Since the angles in a triangle add up to $180^{\circ}$, we have:

$k + x + x = 180^{\circ}$

Also note that the exterior angle to angle $x$ is $83$ degrees.

Therefore,

$x + 83^{\circ} = 180^{\circ}$. Then,

$x = 180^{\circ} - 83^{\circ} = 97^{\circ}$.

So $k + 2x = 180^{\circ}$.

$k + 2(97^{\circ}) = 180^{\circ}$.

$k + 194^{\circ} = 180^{\circ}$.

$k = 180^{\circ} - 194^{\circ} = -14^{\circ}$.

This does not make sense.

The angles k and 83 degrees make a straight line, thus they are supplementary and add up to 180 degrees. Therefore,

$k + 83^{\circ} = 180^{\circ}$

$k = 180^{\circ} - 83^{\circ}$

$k = 97^{\circ}$

The isosceles triangle has two equal angles, say $x$.

Thus the angles in the triangle are $k, x, x$.

The angles in a triangle sum to

1

8

0. So $k + x + x = 180$

$k + 2x = 180$

Since the angle $x$ and 83 degree form an exterior angle. The remote interior angles are $x$ and $k$. Then their sum is 83 degrees, which means $k+x = 83$ degrees.

Thus $2x + k = 180$

$x + k = 83$

Subtract the two equation, we have $x = 180 - 83 = 97$

Then, $k = 83 - 97 = -14$, which makes no sense.

The diagram shows k and 83 as supplementary angles.

3. Final Answer

$k = 97^{\circ}$