So the directions say:

Provide a counterexample to show that each statement is false.
You can use words or make a diagram.

These are the two questions that I'm stuck on:
1. If a line lies in a vertical plane, then the line is vertical
2. If x^2=49, then x=7

I think that the answer to them is this, but I just wanted some verification:
1. If a line is vertical, then the line lies in a vertical plane.
2. x=7 if x^2=49

you stated the converses , but nothing about true/false

1. picture football goalposts as a vertical plane
... the side posts are vertical , the crossbar is not

2. (-7)^2 = 49

To provide counterexamples for these statements, you're looking for cases that disprove the statements by showing a situation where the condition is true but the conclusion is false.

1. The statement is: "If a line lies in a vertical plane, then the line is vertical." To provide a counterexample, you would need to find a line that lies in a vertical plane but is not vertical. One example of such a counterexample is a line lying in a vertical plane, but sloping or slanted, like the line formed by the equation y = x. This line lies in a vertical plane (since it extends infinitely above and below), but it is not vertical.

2. The statement is: "If x^2 = 49, then x = 7." To provide a counterexample, you need to find a value for x that satisfies the condition (x^2 = 49) but does not make the conclusion true (x = 7). One counterexample is x = -7. If you substitute x = -7 into the equation, you get (-7)^2 = 49, which is true. However, x is not equal to 7, so the conclusion does not hold.

Your understanding is correct in the context of finding counterexamples. However, for the first statement, you don't need to make the counterexample converse, as you mentioned. It doesn't necessarily mean that if a line is vertical, it lies in a vertical plane. The original statement asserts that if a line lies in a vertical plane, it must be vertical.

To provide counterexamples to show that each statement is false, you need to find examples where the statements do not hold true. Let's address each question one by one:

1. If a line lies in a vertical plane, then the line is vertical.

Counterexample: Imagine a line that lies in a vertical plane but is not vertical. For instance, consider a diagonal line on a piece of paper. This line lies in a vertical plane (the plane of the paper), but it is not a vertical line since it has an angle with respect to the vertical axis.

2. If x^2 = 49, then x = 7.

Counterexample: Let's find a value of x that satisfies x^2 = 49 but x ≠ 7. For example, when we solve x^2 = 49, we get two possible solutions: x = 7 and x = -7. Therefore, x can take a value of -7 as well, which does not equal 7.

Your understanding of the counterexamples seems correct:

1. The counterexample to show the statement is false is "If a line is vertical, then the line lies in a vertical plane."
2. The counterexample to show the statement is false is "x = 7 if x^2 = 49."

Remember, counterexamples are examples that contradict the given statements.