To tackle the problem, we need to understand a few properties of vertical angles:
1. Vertical angles are the angles opposite each other when two lines intersect. They are always equal.
2. The sum of angles around a point is [tex]\(360^\circ\)[/tex].
3. Any pair of adjacent angles formed by intersecting lines are supplementary, meaning their measures add up to [tex]\(180^\circ\)[/tex].
Now let's analyze the statement: "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."
An acute angle is an angle that is less than [tex]\(90^\circ\)[/tex].
An obtuse angle is an angle that is more than [tex]\(90^\circ\)[/tex].
In order to prove or provide a counterexample for this statement, consider the possible outcomes when two lines intersect:
- If one pair of vertical angles is acute (less than [tex]\(90^\circ\)[/tex]), the adjacent angles will be [tex]\(180^\circ - \text{(acute angle)}\)[/tex], which must be obtuse.
- If one pair of vertical angles is obtuse (more than [tex]\(90^\circ\)[/tex]), the adjacent angles will be [tex]\((180^\circ - \text{obtuse angle})\)[/tex], which must be acute.
However, let's think of specific scenarios or images that could serve as counterexamples. Consider the case where intersecting lines form:
1. Right angles ([tex]\(90^\circ\)[/tex]):
- Here, all four angles are [tex]\(90^\circ\)[/tex].
- Both pairs of vertical angles will be neither acute nor obtuse.
By visualizing this scenario (lines intersecting perpendicularly), we can see that it does not fit the initial claim because neither pair of vertical angles is acute or obtuse; all angles are right angles.
Now, let's select the correct answer based on the analysis:
The best counterexample would be an image that shows intersecting lines forming four [tex]\(90^\circ\)[/tex] angles. If such an image is provided, it would clearly invalidate the statement as there are no acute or obtuse angles formed.
Thus, without the images in front of me, I would infer that the image showing perpendicular lines – where all angles are [tex]\(90^\circ\)[/tex] – is the one that serves as the best counterexample. Assuming the options and descriptions provided correspond accurately:
Image C is likely the best counterexample as it demonstrates intersecting lines forming right angles.
Therefore, the correct answer is:
C.
