Question

determine if the conjectures are true or false. if false, provide a counterexample. 1. all perfect squares are divisible by 2. 2. multiples of 3 are always multiples of 6. 3. the product of a negative and positive number is always a negative. which diagram is a counterexample to the statement, “supplementary angles are never congruent.”

Explanation:

Step1: Analyze statement 1

A perfect - square number can be odd. For example, \(9 = 3\times3\), and 9 is not divisible by 2. So the statement "All perfect squares are divisible by 2" is false. Counter - example: 9.

Step2: Analyze statement 2

Multiples of 3 are not always multiples of 6. A multiple of 3 is of the form \(3n\) where \(n\in\mathbb{Z}\), and a multiple of 6 is of the form \(6m=2\times3m\) where \(m\in\mathbb{Z}\). For example, 3 is a multiple of 3 but not a multiple of 6. So the statement "Multiples of 3 are always multiples of 6" is false. Counter - example: 3.

Step3: Analyze statement 3

The product of a negative and a positive number is \(a\times b\) where \(a\lt0\) and \(b > 0\). By the rule of multiplication of real numbers, \(a\times b=-(|a|\times|b|)\lt0\). So the statement "The product of a negative and positive number is always a negative" is true.

Step4: Analyze counter - example for supplementary angles

Supplementary angles add up to \(180^{\circ}\). Congruent angles have the same measure. If \(\angle1 = 90^{\circ}\) and \(\angle2=90^{\circ}\), they are supplementary and congruent. But if \(\angle1 = 100^{\circ}\) and \(\angle2 = 80^{\circ}\), they are supplementary but not congruent. The diagram where \(\angle1\) and \(\angle2\) are non - congruent supplementary angles (e.g., \(\angle1=100^{\circ}\), \(\angle2 = 80^{\circ}\)) is a counter - example. For the given diagrams, the one where the two angles are clearly non - congruent and supplementary (the last non - right - angled diagram) is the counter - example.

Answer:

1. False, counter - example: 9
2. False, counter - example: 3
3. True
4. The last non - right - angled diagram (where \(\angle1\) and \(\angle2\) are supplementary but not congruent)