Question

1. determine whether the statement is true or false based on the diagram above: planes w and x intersect at line kl. a. true b. false 12. which is a counterexample to this conjecture? \if a number is divisible by 3, then it is also divisible by 6\. a. 6 b. 9 c. 12 d. 18 13. which is the inverse of the statement: \if a shape is a square, then it has four equal sides.\ a. if a shape is not a square, then it does not have four equal sides. b. if a shape has four equal sides, then it is a square. c. if a shape does not have four equal sides, then it is not a square. d. if a shape is a square, then it does not have four equal sides.

Explanation:

1. For question 11:

• Planes intersect in a line. Looking at the diagram (assuming standard plane - intersection concepts in geometry), if planes \(W\) and \(X\) intersect at line \(KL\), and based on the basic property of plane - intersection (two planes intersect in exactly one line).
• Since there is no information in the problem that contradicts this basic geometric property (assuming the diagram is a valid representation of plane - intersection), the statement “Planes \(W\) and \(X\) intersect at line \(KL\)” is True.

1. For question 12:

• A counter - example is a case that makes a conjecture false.
• The conjecture is “If a number is divisible by 3, then it is also divisible by 6”.
• Let's check each option:
• For \(n = 6\), \(6\div3 = 2\) and \(6\div6=1\). So it does not serve as a counter - example.
• For \(n = 9\), \(9\div3 = 3\) (divisible by 3), but \(9\div6=\frac{9}{6}=\frac{3}{2}\) (not divisible by 6).
• For \(n = 12\), \(12\div3 = 4\) and \(12\div6 = 2\). So it does not serve as a counter - example.
• For \(n = 18\), \(18\div3=6\) and \(18\div6 = 3\). So it does not serve as a counter - example.

1. For question 13:

• The original statement is “If \(p\) (a shape is a square), then \(q\) (it has four equal sides)”.
• The inverse of a conditional statement “If \(p\), then \(q\)” is “If not \(p\), then not \(q\)”.
• Option A: “If a shape is not a square (not \(p\)), then it does not have four equal sides (not \(q\))” is in the form of the inverse.
• Option B is the converse (If \(q\), then \(p\)).
• Option C is the contrapositive (If not \(q\), then not \(p\)).
• Option D is a false negation of the original statement.

Answer:

1. A. True
2. B. 9
3. A. If a shape is not a square, then it does not have four equal sides.