Question

independent practice: judge the validity of arguments and give counterexamples to disprove statements: if x is an integer, then -x is positive. if ∠2 and ∠3 are supplementary angles, then ∠2 and ∠3 form a linear pair. if you have three points a, b, and c, then a, b, and c are non - collinear. if in △abc, (ab)^2+(bc)^2=(ac)^2, then △abc is a right triangle.

Explanation:

Step1: Analyze the first statement

If \(x = 5\) (a positive - integer), then \(-x=-5\) which is negative. So the statement "If \(x\) is an integer, then \(-x\) is positive" is false.

Step2: Analyze the second statement

Two supplementary angles \(\angle2\) and \(\angle3\) do not necessarily form a linear - pair. For example, two non - adjacent angles in a parallelogram can be supplementary but not form a linear pair. So the statement "If \(\angle2\) and \(\angle3\) are supplementary angles, then \(\angle2\) and \(\angle3\) form a linear pair" is false.

Step3: Analyze the third statement

If \(A\), \(B\), and \(C\) are on the same straight line, they are collinear. For example, if \(A\), \(B\), and \(C\) are points on a number line, they are collinear. So the statement "If you have three points \(A\), \(B\), and \(C\), then \(A\), \(B\), and \(C\) are non - collinear" is false.

Step4: Analyze the fourth statement

This is the Pythagorean theorem. If \((AB)^{2}+(BC)^{2}=(AC)^{2}\) in \(\triangle ABC\), then \(\triangle ABC\) is a right - triangle with the right angle opposite the side \(AC\). This statement is true.

Answer:

The first statement is false (counter - example: \(x = 5\)).
The second statement is false (counter - example: non - adjacent supplementary angles in a parallelogram).
The third statement is false (counter - example: points on a number line).
The fourth statement is true.