Answer :
To determine whether 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" is true or false, we'll examine some key points about vertical angles and intersecting lines.
1. Understanding Vertical Angles:
- When two lines intersect, they form two pairs of vertical (opposite) angles.
- Vertical angles are congruent, meaning each pair of vertical angles is equal in measure.
2. Relationship Between Angles:
- The two angles that are adjacent (next to each other) to a vertical angle are supplementary to it. This means the sum of these two angles is 180 degrees.
- For example, if one angle formed by the intersecting lines is represented as [tex]\( \theta \)[/tex], another angle adjacent to it would be [tex]\( 180^\circ - \theta \)[/tex].
With this understanding, let's explore various cases:
### Case 1: Both Pairs of Angles are Acute and Obtuse
- Suppose two angles are [tex]\( \theta \)[/tex] and [tex]\( 180^\circ - \theta \)[/tex].
- If [tex]\( 0^\circ < \theta < 90^\circ \)[/tex], then the angle [tex]\( 180^\circ - \theta \)[/tex] would be obtuse (greater than 90 degrees). In this case, one pair is acute, and the other pair is obtuse.
- Similarly, if [tex]\( 90^\circ < \theta < 180^\circ \)[/tex], then the angle [tex]\( 180^\circ - \theta \)[/tex] would be acute. This scenario would also support the statement.
### Case 2: Both Pairs of Angles are Right Angles
- If the intersecting lines are perpendicular, each angle formed will be [tex]\( 90^\circ \)[/tex].
- Both pairs of vertical angles will be [tex]\( 90^\circ \)[/tex], thus neither acute nor obtuse, contradicting the statement that one pair would be acute and one pair would be obtuse.
### Conclusion:
The statement can be disproved by providing a scenario where both pairs of vertical angles are right angles. Such a counterexample proves that it is not necessary for one pair to be acute and one pair to be obtuse.
Given this counterexample, the correct image referenced needs to show intersecting lines forming right angles (90 degrees each). This scenario provides evidence against the statement. Each image would need to be examined to verify which represents perpendicular intersections.
Since no specific images are provided here, the important conclusion is that an image where intersecting lines form exactly 90-degree angles each would be the correct counterexample for proving the statement false.
1. Understanding Vertical Angles:
- When two lines intersect, they form two pairs of vertical (opposite) angles.
- Vertical angles are congruent, meaning each pair of vertical angles is equal in measure.
2. Relationship Between Angles:
- The two angles that are adjacent (next to each other) to a vertical angle are supplementary to it. This means the sum of these two angles is 180 degrees.
- For example, if one angle formed by the intersecting lines is represented as [tex]\( \theta \)[/tex], another angle adjacent to it would be [tex]\( 180^\circ - \theta \)[/tex].
With this understanding, let's explore various cases:
### Case 1: Both Pairs of Angles are Acute and Obtuse
- Suppose two angles are [tex]\( \theta \)[/tex] and [tex]\( 180^\circ - \theta \)[/tex].
- If [tex]\( 0^\circ < \theta < 90^\circ \)[/tex], then the angle [tex]\( 180^\circ - \theta \)[/tex] would be obtuse (greater than 90 degrees). In this case, one pair is acute, and the other pair is obtuse.
- Similarly, if [tex]\( 90^\circ < \theta < 180^\circ \)[/tex], then the angle [tex]\( 180^\circ - \theta \)[/tex] would be acute. This scenario would also support the statement.
### Case 2: Both Pairs of Angles are Right Angles
- If the intersecting lines are perpendicular, each angle formed will be [tex]\( 90^\circ \)[/tex].
- Both pairs of vertical angles will be [tex]\( 90^\circ \)[/tex], thus neither acute nor obtuse, contradicting the statement that one pair would be acute and one pair would be obtuse.
### Conclusion:
The statement can be disproved by providing a scenario where both pairs of vertical angles are right angles. Such a counterexample proves that it is not necessary for one pair to be acute and one pair to be obtuse.
Given this counterexample, the correct image referenced needs to show intersecting lines forming right angles (90 degrees each). This scenario provides evidence against the statement. Each image would need to be examined to verify which represents perpendicular intersections.
Since no specific images are provided here, the important conclusion is that an image where intersecting lines form exactly 90-degree angles each would be the correct counterexample for proving the statement false.
