Answer :
To address this geometry question and prove or disprove the given statement, let's follow a step-by-step explanation.
Given 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.
Step-by-step Explanation:
1. Understanding Vertical Angles:
- When two lines intersect, they form two pairs of opposite (vertical) angles. Vertical angles are congruent, meaning they have equal measure. For example, if two lines intersect at point [tex]\( O \)[/tex], then the angles [tex]\( \angle AOC \)[/tex] and [tex]\( \angle BOD \)[/tex] are one pair of vertical angles, and [tex]\( \angle AOD \)[/tex] and [tex]\( \angle BOC \)[/tex] are the other pair.
2. Acute Angles:
- An acute angle is one that is less than 90 degrees.
3. Obtuse Angles:
- An obtuse angle is one that is more than 90 degrees but less than 180 degrees.
4. Considering Possible Configurations:
- Let's consider the four angles formed by the intersecting lines. Since the sum of angles around a point is 360 degrees, we know that:
[tex]\[ \angle AOC + \angle COB + \angle BOD + \angle DOA = 360^\circ \][/tex]
- Vertical angles are equal, so we have:
[tex]\[ \angle AOC = \angle BOD \quad \text{and} \quad \angle AOD = \angle BOC \][/tex]
5. Exploring the Given Statement:
- According to the given statement, if one of the pairs of vertical angles is acute, the other pair should be obtuse. This is possible because the sum of an acute angle and its adjacent angle must be 180 degrees (they form a straight line). For example, if [tex]\( \angle AOC \)[/tex] is 60 degrees (acute), then [tex]\( \angle AOD \)[/tex] would be 120 degrees (obtuse).
6. Identifying the Counterexample:
- The given statement implies that it is always true that one pair is acute and the other pair is obtuse. To disprove this, we need to find a scenario where either both pairs are right angles (each pair is 90 degrees), or another configuration exists that violates the statement.
- Imagine two lines intersecting perpendicularly. In this case, all four angles are 90 degrees:
[tex]\[ \angle AOC = \angle BOD = 90^\circ \quad \text{and} \quad \angle AOD = \angle BOC = 90^\circ \][/tex]
This configuration shows that the given statement is not always true. All four angles are right angles, and neither pair of vertical angles includes an acute or obtuse angle. Thus, the best counterexample would be an image showing two lines intersecting perpendicularly, forming four right angles.
Conclusion:
To provide the correct counterexample image to disprove the given statement, look for an image where two lines intersect at right angles, forming four 90-degree angles.
Given 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.
Step-by-step Explanation:
1. Understanding Vertical Angles:
- When two lines intersect, they form two pairs of opposite (vertical) angles. Vertical angles are congruent, meaning they have equal measure. For example, if two lines intersect at point [tex]\( O \)[/tex], then the angles [tex]\( \angle AOC \)[/tex] and [tex]\( \angle BOD \)[/tex] are one pair of vertical angles, and [tex]\( \angle AOD \)[/tex] and [tex]\( \angle BOC \)[/tex] are the other pair.
2. Acute Angles:
- An acute angle is one that is less than 90 degrees.
3. Obtuse Angles:
- An obtuse angle is one that is more than 90 degrees but less than 180 degrees.
4. Considering Possible Configurations:
- Let's consider the four angles formed by the intersecting lines. Since the sum of angles around a point is 360 degrees, we know that:
[tex]\[ \angle AOC + \angle COB + \angle BOD + \angle DOA = 360^\circ \][/tex]
- Vertical angles are equal, so we have:
[tex]\[ \angle AOC = \angle BOD \quad \text{and} \quad \angle AOD = \angle BOC \][/tex]
5. Exploring the Given Statement:
- According to the given statement, if one of the pairs of vertical angles is acute, the other pair should be obtuse. This is possible because the sum of an acute angle and its adjacent angle must be 180 degrees (they form a straight line). For example, if [tex]\( \angle AOC \)[/tex] is 60 degrees (acute), then [tex]\( \angle AOD \)[/tex] would be 120 degrees (obtuse).
6. Identifying the Counterexample:
- The given statement implies that it is always true that one pair is acute and the other pair is obtuse. To disprove this, we need to find a scenario where either both pairs are right angles (each pair is 90 degrees), or another configuration exists that violates the statement.
- Imagine two lines intersecting perpendicularly. In this case, all four angles are 90 degrees:
[tex]\[ \angle AOC = \angle BOD = 90^\circ \quad \text{and} \quad \angle AOD = \angle BOC = 90^\circ \][/tex]
This configuration shows that the given statement is not always true. All four angles are right angles, and neither pair of vertical angles includes an acute or obtuse angle. Thus, the best counterexample would be an image showing two lines intersecting perpendicularly, forming four right angles.
Conclusion:
To provide the correct counterexample image to disprove the given statement, look for an image where two lines intersect at right angles, forming four 90-degree angles.
