Let's analyze the statements provided and determine which one is not true:

1. Statement A: "Given any two distinct lines in the Cartesian plane, the two lines will either be parallel or perpendicular."

   • This is incorrect. Two lines in a plane can intersect at an angle that is not 90 degrees, meaning they are neither parallel nor perpendicular. Hence, this statement is false.

2. Statement B: "If two distinct nonvertical lines are parallel, then the two lines must have the same slope."

   • This is true. Parallel lines in the Cartesian plane have equal slopes.

3. Statement C: "Given any two distinct lines in the Cartesian plane, the two lines will either intersect or they will be parallel."

   • This is true. Distinct lines that are not parallel will always intersect at some point.

4. Statement D: "The ratios of the vertical rise to the horizontal run of any two distinct nonvertical parallel lines must be equal."

   • This is true. The ratio of the rise to the run represents the slope of the line, and parallel lines have the same slope.

Conclusion:

The incorrect statement is A.

Would you like more details or have any questions about this explanation?

Here are some related questions to deepen your understanding:

1. How do you determine the slope of a line given two points?
2. What is the general form of a line in the Cartesian plane?
3. How can you prove if two lines are perpendicular using their slopes?
4. How do you find the point of intersection between two lines?
5. What are the conditions for two lines to be coincident?

Tip: For two lines to be perpendicular, the product of their slopes should equal $-1$.