Question Analysis:
The question asks us to determine the relationship between two angles when their measures sum up to that of a straight angle, and one of the angles is acute. We need to choose the correct option from the given choices.
Given Information:
- The sum of the measures of two angles is equal to that of a straight angle.
- One of the angles is acute.

Solution:
Understanding the Terminology:
- Straight Angle: A straight angle measures exactly 180 degrees.
- Acute Angle: An acute angle measures less than 90 degrees.
- Obtuse Angle: An obtuse angle measures more than 90 degrees but less than 180 degrees.
- Right Angle: A right angle measures exactly 90 degrees.

Using the Given Information:
Since the sum of the measures of the two angles is equal to that of a straight angle (180 degrees), we can write the equation:

Measure of Angle 1 + Measure of Angle 2 = 180

Now, let's consider two cases based on the given information:

Case 1: One of the angles is acute:
Let's assume Angle 1 is acute.
- The measure of Angle 1 is less than 90 degrees.
- The measure of Angle 2 can be found by subtracting the measure of Angle 1 from 180, as shown below:
Measure of Angle 2 = 180 - Measure of Angle 1

Case 2: One of the angles is not acute:
If one of the angles is not acute, it means it can be either a right angle, an obtuse angle, or a straight angle.
- If one of the angles is a right angle (90 degrees), then the other angle must be a straight angle (180 degrees) in order for their sum to be equal to 180 degrees.
- If one of the angles is an obtuse angle (more than 90 degrees but less than 180 degrees), then the other angle must be an acute angle in order for their sum to be equal to 180 degrees.

Conclusion:
Based on the given information and the analysis above, we can conclude that if one of the angles is acute, then the other angle must be obtuse. Therefore, the correct answer is option A, "obtuse".