Intersecting Lines and Parallel Lines

Introduction:
When dealing with intersecting lines and parallel lines, it is important to understand the properties and relationships between these lines. In this case, we have two intersecting lines, labeled as l and l. We are given that line l is parallel to line P, and line M is parallel to line Q. The task is to prove that lines P and Q also intersect.

Proof:
To prove that lines P and Q intersect, we can use the properties of parallel lines and intersecting lines.

Parallel Lines:
Parallel lines are lines that never intersect. If two lines are parallel, they will always remain equidistant from each other and will never cross paths.

Intersecting Lines:
Intersecting lines are lines that cross each other at a single point. If two lines intersect, they share a common point, which is known as the point of intersection.

Given Information:
1. Line l is parallel to line P.
2. Line M is parallel to line Q.
3. Lines l and l intersect.

Conclusion:
Based on the given information, we can conclude that lines P and Q intersect as well. Here's the reasoning:

1. Since lines l and l intersect, they share a common point of intersection.
2. Line l is parallel to line P, which means that line P will never intersect with line l.
3. Line M is parallel to line Q, which means that line Q will never intersect with line M.
4. However, since lines P and Q are not parallel to each other, they have the potential to intersect.
5. Therefore, the only possibility is that lines P and Q intersect at the same point where lines l and l intersect.

Conclusion:
In conclusion, based on the given information and the properties of parallel lines and intersecting lines, we can confidently say that lines P and Q intersect. The fact that lines l and l intersect, and that lines P and Q are not parallel to each other, ensures that they share a common point of intersection.