To find the sum of the measures of the interior angles at vertex I and vertex J of triangle HIJ, we can use the properties of triangles and exterior angles.
The exterior angle at vertex H is given as 38 degrees. By the Exterior Angle Theorem, we know that the measure of an exterior angle is equal to the sum of the measures of the two opposite interior angles. Thus, the exterior angle at H can be expressed as:
\[ \text{Exterior angle at H} = \text{Interior angle at I} + \text{Interior angle at J} \]
Let the interior angles at vertices I and J be represented as \( \angle I \) and \( \angle J \). Therefore, we can write:
\[ \angle I + \angle J = 38^\circ \]
However, we are not looking for the individual measures of angles I and J, but rather their sum.
Using the knowledge that the sum of the interior angles of any triangle is always 180 degrees, we can express this total as:
\[ \angle H + \angle I + \angle J = 180^\circ \]
Since \( \angle H \) is the angle opposite to the exterior angle at H, we can relate it to the sum of angles I and J via:
\[ \angle H = 180^\circ - ( \angle I + \angle J ) \]
Equating it to our previously established value for the exterior angle gives us:
Since \( \angle I + \angle J = 38^\circ \), substituting \( \angle H = 180^\circ - 38^\circ \):
\[ \angle H = 142^\circ \]
Thus, the sum of the interior angles at vertex I and vertex J is:
\[ \angle I + \angle J = 180^\circ - \angle H = 180^\circ - 142^\circ = 38^\circ \]
From the calculations, the sum of the measures of the interior angles at vertex I and vertex J is therefore:
\[ \text{Answer: } 142^\circ \]
So the correct response is 142°.