Question
Lines l and m are horizontal and are intersected by lines p and q. At the intersection of lines l and p, the uppercase left angle is 130 degrees. At the intersection of lines q and l, the bottom right angle is 130 degrees. At the intersection of lines m and q, the uppercase right angle is 80 degrees.
Which lines are parallel? Justify your answer.
Which lines are parallel? Justify your answer.
Answers
GPT-4o mini
To determine which lines are parallel, let's analyze the information given about the angles formed by the intersections of the lines.
1. **At the intersection of lines l and p**, there is an angle of 130 degrees in the upper left corner. Since lines l and p are intersecting, the angle directly below it (the lower right angle) must be supplementary to 130 degrees, which means it is \(180 - 130 = 50\) degrees.
2. **At the intersection of lines l and q**, the angle in the bottom right corner is given as 130 degrees. The angle directly above it (the upper left angle) must also be supplementary to 130 degrees, therefore it is also 50 degrees.
3. **At the intersection of lines m and q**, the angle in the upper right corner is given as 80 degrees. The angle directly below it (the lower left angle) would be supplementary to 80 degrees, which means it is \(180 - 80 = 100\) degrees.
Now, let's summarize the angles at the intersections involving horizontal lines l and m:
- For lines l and p: We have one angle of 130 degrees and the vertical angle of 50 degrees.
- For lines l and q: We have one angle of 130 degrees and the vertical angle of 50 degrees, which confirms that the angle measures are consistent.
- For lines m and q: We encountered an angle of 80 degrees and therefore the angle opposite to it (lower left) is 100 degrees.
To check if any lines are parallel, we can apply the concept of angle relationships:
- If two lines are crossed by another line (a transversal) and the alternate interior angles are equal, the two lines are parallel.
In this case:
- Looking at lines l and m which are both horizontal and the angles related to lines p and q, we find that the corresponding angles at l and q (130 degrees) match with the supplementary angles formed at the intersection of line m with line q. The 130-degree angle at line q is equal to the 50-degree angle found at the intersection of l with p.
Thus we hypothesize:
- Since the angles created by line q with both lines l and m are related through these corresponding angles (both have a parallel nature due to equal angle measures), we can conclude:
- **Lines l and m are parallel.**
### Justification:
Lines l and m are confirmed as parallel based on corresponding angles created by transversals p and q that meet at these lines: both lines have angles of 130 degrees at one intersection and reflect consistent angle measures at the given intersections. Additionally, if they are horizontal, they follow the norm of parallelism as horizontal lines inherently do not intersect if maintained parallel positions.
1. **At the intersection of lines l and p**, there is an angle of 130 degrees in the upper left corner. Since lines l and p are intersecting, the angle directly below it (the lower right angle) must be supplementary to 130 degrees, which means it is \(180 - 130 = 50\) degrees.
2. **At the intersection of lines l and q**, the angle in the bottom right corner is given as 130 degrees. The angle directly above it (the upper left angle) must also be supplementary to 130 degrees, therefore it is also 50 degrees.
3. **At the intersection of lines m and q**, the angle in the upper right corner is given as 80 degrees. The angle directly below it (the lower left angle) would be supplementary to 80 degrees, which means it is \(180 - 80 = 100\) degrees.
Now, let's summarize the angles at the intersections involving horizontal lines l and m:
- For lines l and p: We have one angle of 130 degrees and the vertical angle of 50 degrees.
- For lines l and q: We have one angle of 130 degrees and the vertical angle of 50 degrees, which confirms that the angle measures are consistent.
- For lines m and q: We encountered an angle of 80 degrees and therefore the angle opposite to it (lower left) is 100 degrees.
To check if any lines are parallel, we can apply the concept of angle relationships:
- If two lines are crossed by another line (a transversal) and the alternate interior angles are equal, the two lines are parallel.
In this case:
- Looking at lines l and m which are both horizontal and the angles related to lines p and q, we find that the corresponding angles at l and q (130 degrees) match with the supplementary angles formed at the intersection of line m with line q. The 130-degree angle at line q is equal to the 50-degree angle found at the intersection of l with p.
Thus we hypothesize:
- Since the angles created by line q with both lines l and m are related through these corresponding angles (both have a parallel nature due to equal angle measures), we can conclude:
- **Lines l and m are parallel.**
### Justification:
Lines l and m are confirmed as parallel based on corresponding angles created by transversals p and q that meet at these lines: both lines have angles of 130 degrees at one intersection and reflect consistent angle measures at the given intersections. Additionally, if they are horizontal, they follow the norm of parallelism as horizontal lines inherently do not intersect if maintained parallel positions.