Question

determine whether the conjecture is true or false. if false, provide a counterexample. 11. the sum of any two consecutive integers is always odd. 12. the product of two numbers is always larger than either number. 13. all four - sided figures are rectangles. 14. if the area of a rectangle is 6 m², then the dimensions must be 2 meters by 3 meters. 15. the difference between two negative numbers is always negative. 16. vertical angles are never complementary angles. 17. all states have at least 2 syllables.

Explanation:

Step1: Analyze consecutive integers

Let the first integer be $n$, the next consecutive integer is $n + 1$. Their sum is $n+(n + 1)=2n + 1$. Since $2n$ is even for any integer $n$, $2n+1$ is odd. So the statement "The sum of any two consecutive integers is always odd" is true.

Step2: Analyze product of numbers

Consider $0.5\times0.2 = 0.1$. Here, $0.1<0.5$ and $0.1<0.2$. So the statement "The product of two numbers is always larger than either number" is false. Counter - example: $0.5$ and $0.2$.

Step3: Analyze four - sided figures

A rhombus is a four - sided figure but not a rectangle (unless it has right angles). So the statement "All four - sided figures are rectangles" is false. Counter - example: a non - rectangular rhombus.

Step4: Analyze rectangle dimensions

If the area of a rectangle $A = lw=6$ $m^{2}$, it could also be $1$ meter by $6$ meters. So the statement "If the area of a rectangle is $6$ $m^{2}$, then the dimensions must be 2 meters by 3 meters" is false. Counter - example: 1 meter by 6 meters.

Step5: Analyze difference of negative numbers

Consider $-2-(-3)=-2 + 3=1$. So the statement "The difference between two negative numbers is always negative" is false. Counter - example: $-2$ and $-3$.

Step6: Analyze vertical angles

If two lines intersect at right angles, vertical angles are complementary (each is 45 degrees). So the statement "Vertical angles are never complementary angles" is false. Counter - example: two lines intersecting at right angles.

Step7: Analyze state names

The state "Maine" has 1 syllable. So the statement "All states have at least 2 syllables" is false. Counter - example: Maine.

Answer:

1. True
2. False, Counter - example: 0.5 and 0.2
3. False, Counter - example: a non - rectangular rhombus
4. False, Counter - example: 1 meter by 6 meters
5. False, Counter - example: -2 and -3
6. False, Counter - example: two lines intersecting at right angles
7. False, Counter - example: Maine