To address the question of providing a counterexample to the statement that if two intersecting lines form two pairs of vertical angles, one pair of angles will be acute and one pair of angles will be obtuse, we need to first understand the basic properties of vertical angles.
### Key Points:
1. Vertical Angles: When two lines intersect, they form two pairs of vertical (opposite) angles.
2. Properties of Vertical Angles:
- Vertical angles are always equal.
- Adjacent angles formed by two intersecting lines are supplementary, meaning that the sum of their measures is 180 degrees.
Given these properties, let's consider the types of angles that can form when lines intersect at different angles:
- If the two intersecting lines form right angles (90 degrees each), both pairs of vertical angles would be right angles.
- Right angles are neither acute (less than 90 degrees) nor obtuse (greater than 90 degrees).
Therefore, if the intersecting lines form right angles, it disproves the statement as neither angle in the pairs is acute or obtuse.
### Counterexample:
The best counterexample demonstrates a pair of intersecting lines forming right angles. In such a scenario:
- Each angle between the intersecting lines would be exactly 90 degrees.
- All four angles formed at the intersection would be right angles, and therefore, no vertical angle would be acute or obtuse.
### Conclusion:
The best counterexample out of the options provided would be the one that indicates intersecting lines forming right angles.
Correct Answer: D. [tex]\(\uparrow \uparrow\)[/tex]
