Question
Horizontal lines e and f are intersected by lines a and b. At the intersection of lines a and e, the uppercase left angle is 75 degrees. At the intersection of lines b and e, the uppercase right angle is 115 degrees. At the intersection of lines a and f, the bottom right angle is 75 degrees.
Which lines are parallel? Justify your answer.
Lines a and b are parallel because their alternate exterior angles are congruent.
Lines a and b are parallel because their same side exterior angles are supplementary.
Lines e and f are parallel because their alternate exterior angles are congruent.
Lines e and f are parallel because their same side exterior angles are congruent.
Which lines are parallel? Justify your answer.
Lines a and b are parallel because their alternate exterior angles are congruent.
Lines a and b are parallel because their same side exterior angles are supplementary.
Lines e and f are parallel because their alternate exterior angles are congruent.
Lines e and f are parallel because their same side exterior angles are congruent.
Answers
GPT-4o mini
To determine which lines are parallel and to justify the conclusions based on the angles provided, let's analyze the angles at the intersections of the lines.
1. **At the intersection of lines a and e**, the uppercase left angle is 75 degrees (let's call this angle A). The angle directly opposite angle A (the corresponding angle on the right side of line a) is also 75 degrees due to the property of vertical angles.
2. **At the intersection of lines b and e**, the uppercase right angle is 115 degrees (let's call this angle B). Since lines e and b are horizontal, the angle on the left (corresponding angle) at the intersection with line e is also 115 degrees.
3. **At the intersection of lines a and f**, the bottom right angle is 75 degrees (let's call this angle C). Its alternate angle at the point of intersection with line e (to the top left of line a) is also 75 degrees.
Now, let's analyze parallelism:
- **Lines a and b**:
- Angle A (75 degrees) and Angle B (the corresponding angle on line b which is also the angle across from 115 degrees) are alternate angles when considering line e. Since 75 degrees is not equal to 115 degrees, lines a and b are not parallel.
- **Lines e and f**:
- The angles formed at intersections with lines a and f are both 75 degrees. Thus, both lines have alternate interior angles that are equal (i.e., the angle across from 75 degrees at the intersection of line a and e with line f). By the Alternate Interior Angles Theorem, since one pair of alternate interior angles are equal, lines e and f are parallel.
Thus, the correct conclusion is:
**Lines e and f are parallel because their alternate exterior angles are congruent.**
1. **At the intersection of lines a and e**, the uppercase left angle is 75 degrees (let's call this angle A). The angle directly opposite angle A (the corresponding angle on the right side of line a) is also 75 degrees due to the property of vertical angles.
2. **At the intersection of lines b and e**, the uppercase right angle is 115 degrees (let's call this angle B). Since lines e and b are horizontal, the angle on the left (corresponding angle) at the intersection with line e is also 115 degrees.
3. **At the intersection of lines a and f**, the bottom right angle is 75 degrees (let's call this angle C). Its alternate angle at the point of intersection with line e (to the top left of line a) is also 75 degrees.
Now, let's analyze parallelism:
- **Lines a and b**:
- Angle A (75 degrees) and Angle B (the corresponding angle on line b which is also the angle across from 115 degrees) are alternate angles when considering line e. Since 75 degrees is not equal to 115 degrees, lines a and b are not parallel.
- **Lines e and f**:
- The angles formed at intersections with lines a and f are both 75 degrees. Thus, both lines have alternate interior angles that are equal (i.e., the angle across from 75 degrees at the intersection of line a and e with line f). By the Alternate Interior Angles Theorem, since one pair of alternate interior angles are equal, lines e and f are parallel.
Thus, the correct conclusion is:
**Lines e and f are parallel because their alternate exterior angles are congruent.**