In a triangle, the sum of the interior angles is always 180 degrees.
In triangle HIJ, if we know that there is an exterior angle at vertex H measuring 38 degrees, we can use the property that an exterior angle is equal to the sum of the two opposite interior angles. The interior angles at vertices I and J will be equal to the exterior angle at vertex H plus the interior angle at vertex H.
Let's denote the angles at vertices I, J, and H as ∠I, ∠J, and ∠H respectively. According to the exterior angle theorem:
\[ \angle H + \angle I + \angle J = 180^\circ \quad (\text{sum of angles in a triangle}) \] \[ \angle H + \angle J = 38^\circ \quad (\text{from the exterior angle definition}) \]
Now we can express the relationship as follows:
\[ \angle H = 180^\circ - (\angle I + \angle J) = 38^\circ \]
This means that:
\[ \angle I + \angle J = 180^\circ - 38^\circ = 142^\circ \]
Therefore, the sum of the measures of the interior angles at vertex I and vertex J is 142 degrees.
So, the correct response is:
142°.