In a triangle, the sum of the measures of the three interior angles is always 180 degrees.
Given that there is an exterior angle of 38 degrees at vertex H, we can use the exterior angle theorem, which states that the measure of an exterior angle is equal to the sum of the measures of the two opposite interior angles.
Thus, we have:
\[ \text{Exterior angle at H} = \text{Interior angle at I} + \text{Interior angle at J} \] \[ 38° = \text{Angle I} + \text{Angle J} \]
We can call the interior angles at I and J as \( x \) and \( y \) respectively, so:
\[ x + y = 38° \]
Now, since the sum of all interior angles in triangle HIJ is 180 degrees, we have:
\[ x + y + \text{Angle H} = 180° \]
Here, \(\text{Angle H} = 180° - 38° = 142°\) (because angle H is the interior angle opposite the exterior angle).
To find the sum of the measures of the angles at vertex I and vertex J, we can use:
\[ x + y = 180° - 142° = 38° \]
The question asks for the sum of the measures of the interior angles at vertex I and vertex J, and from our calculations, we have:
The sum of the measures of the interior angles at vertex I and vertex J is 142°.
Thus, the correct answer is:
142°