Answer :
To address this question, let's first examine what it means for angles to be "vertical angles." Vertical angles are the angles that are opposite each other when two lines intersect. They are formed by the same two lines and equal each other in measure.
The claim states: "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."
To analyze this statement, let's consider the following definitions:
- Acute angles are angles that measure less than 90 degrees.
- Obtuse angles are angles that measure more than 90 degrees but less than 180 degrees.
When two lines intersect, they form four angles. Let's denote these angles as [tex]\( \angle 1, \angle 2, \angle 3,\)[/tex] and [tex]\( \angle 4 \)[/tex], where:
- [tex]\( \angle 1 \)[/tex] and [tex]\( \angle 3 \)[/tex] are vertical angles (opposite and equal).
- [tex]\( \angle 2 \)[/tex] and [tex]\( \angle 4 \)[/tex] are vertical angles (opposite and equal).
According to the properties of intersecting lines:
- [tex]\( \angle 1 + \angle 2 = 180^\circ \)[/tex] (since they form a linear pair).
- Therefore, [tex]\( \angle 3 \)[/tex] (which is equal to [tex]\( \angle 1 \)[/tex]) and [tex]\( \angle 4 \)[/tex] (which is equal to [tex]\( \angle 2 \)[/tex]) should also sum up to 180 degrees.
From the above, we can draw the conclusion:
- If [tex]\( \angle 1 \)[/tex] is acute (less than 90 degrees), then [tex]\( \angle 2 \)[/tex] (and consequently [tex]\( \angle 4 \)[/tex]) has to be obtuse (greater than 90 degrees but less than 180 degrees).
- Conversely, if [tex]\( \angle 1 \)[/tex] is obtuse, then [tex]\( \angle 2 \)[/tex] (and [tex]\( \angle 4 \)[/tex]) has to be acute.
Hence, the assertion in the claim seems plausible based on the properties of vertical angles and linear pairs. However, we must find the correct counterexample to show that this claim is not always true.
For a counterexample:
- Imagine a situation where the intersecting lines form right angles with each other.
- In this specific case, all four angles formed are exactly 90 degrees each, which are neither acute nor obtuse but right angles.
This scenario exemplifies the counterexample that adequately disproves the claim. Thus, we need to choose an image that depicts two intersecting lines forming four right angles.
Given the constraints of this problem in the text form and assuming an image that accurately portrays intersecting lines forming four right angles, the correct counterexample can be identified.
Therefore, the correct image that provides the best counterexample should depict:
- Two intersecting lines forming four 90-degree angles.
So, for the selection among the options (A, B, and C), you would typically need to look at the images provided to pick the one showing two lines intersecting at right angles.
However, as the specifics of the images are not available here, you must select the one that corresponds to the scenario described.
The claim states: "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."
To analyze this statement, let's consider the following definitions:
- Acute angles are angles that measure less than 90 degrees.
- Obtuse angles are angles that measure more than 90 degrees but less than 180 degrees.
When two lines intersect, they form four angles. Let's denote these angles as [tex]\( \angle 1, \angle 2, \angle 3,\)[/tex] and [tex]\( \angle 4 \)[/tex], where:
- [tex]\( \angle 1 \)[/tex] and [tex]\( \angle 3 \)[/tex] are vertical angles (opposite and equal).
- [tex]\( \angle 2 \)[/tex] and [tex]\( \angle 4 \)[/tex] are vertical angles (opposite and equal).
According to the properties of intersecting lines:
- [tex]\( \angle 1 + \angle 2 = 180^\circ \)[/tex] (since they form a linear pair).
- Therefore, [tex]\( \angle 3 \)[/tex] (which is equal to [tex]\( \angle 1 \)[/tex]) and [tex]\( \angle 4 \)[/tex] (which is equal to [tex]\( \angle 2 \)[/tex]) should also sum up to 180 degrees.
From the above, we can draw the conclusion:
- If [tex]\( \angle 1 \)[/tex] is acute (less than 90 degrees), then [tex]\( \angle 2 \)[/tex] (and consequently [tex]\( \angle 4 \)[/tex]) has to be obtuse (greater than 90 degrees but less than 180 degrees).
- Conversely, if [tex]\( \angle 1 \)[/tex] is obtuse, then [tex]\( \angle 2 \)[/tex] (and [tex]\( \angle 4 \)[/tex]) has to be acute.
Hence, the assertion in the claim seems plausible based on the properties of vertical angles and linear pairs. However, we must find the correct counterexample to show that this claim is not always true.
For a counterexample:
- Imagine a situation where the intersecting lines form right angles with each other.
- In this specific case, all four angles formed are exactly 90 degrees each, which are neither acute nor obtuse but right angles.
This scenario exemplifies the counterexample that adequately disproves the claim. Thus, we need to choose an image that depicts two intersecting lines forming four right angles.
Given the constraints of this problem in the text form and assuming an image that accurately portrays intersecting lines forming four right angles, the correct counterexample can be identified.
Therefore, the correct image that provides the best counterexample should depict:
- Two intersecting lines forming four 90-degree angles.
So, for the selection among the options (A, B, and C), you would typically need to look at the images provided to pick the one showing two lines intersecting at right angles.
However, as the specifics of the images are not available here, you must select the one that corresponds to the scenario described.
