Geometry Questions

Question

Show the conjecture is false by finding a counterexample.

1. If the product of two numbers is positive, then the two numbers must

be positive.

2. The sum of any two prime numbers is always even.

For the given statement, write the if-then form, the converse,

the inverse, and the contrapositive.

3.The measure of a straight angle is 180°

4. A quadrilateral has four sides.

Make a valid conclusion in the situation

5.Ifx > 5, then x + 7 > 1 I. The value ofx is .8

6. If you save at least $150, then you can purchase a bicycle on sale.

You have saved $135.

I need help with these 6 geometry questions, i just need them to be solved so i can study from them and take notes

Answer 1

For the given statement, write the if-then form, the converse,

the inverse, and the contrapositive.

For these problems you need to know the following:

Conditional: If-then statement

Converse: Exchange the hypothesis and conclusion of your conditional (reverse order)

Inverse: Negate your conditional statement (Not statement)

Contrapositive: Both reverse and negate your Conditional

3.The measure of a straight angle is 180°

Conditional: If an angle is a straight angle, then its measure is 180.

Converse: If an angle has a measure of 180, then its a straight angle

Inverse: If an angle is not a straight angle, then its measure is not 180.

Contrapositive: If an angle doesn't have a measure of 180, then its not a straight angle.

4. A quadrilateral has four sides.

Conditional: If a figure is a quadrilateral, then it has four sides.

Converse: If a figure has four sides, then its a quadrilateral.

Inverse: If a figure is not a quadrilateral, then it doesn't have four sides.

Contrapositive: If a figure does not have four sides, then its not a quadrilateral.

Answer 2

counterexample to 1: -2 x -2 = +4

counterexample to 2: 2+3 = 5

2 and 3 are prime, 5 is odd, not even

3: a straight angle may be 180 or 180 + n360 where n is any integer

a straight angle could be -180 degrees, + 180+360 = 540 degrees

or an infinite number of other possibilities

4: Hard to think of a possible counter example, as quadralateral means 4 sides

Maybe a degenerate form of a quadralateral where 2 sides are of zero length

or even all four sides are of zero length. Then the figure collapses to a line or a point

5: x>5, x+7>11 is equivalent to x>4. True if x>5 then it's also >4, but x could equal other values than 8.

x=6, 7, 9, 10, 11, or 12

6: If you save at least $150, then you have saved at least $135. Let S= savings. If S>150 then S>135.

Not clear what the bicycle is about. In microeconomics, this might be a question involving opportunity cost which is different from the mathematical cost. For example, you buy the bicycle, but by doing so, it cost more than the price of the bicycle, as you gave up other opportunities, such as buying an airplane ticket for a vacation. The bicycle is on sale, so apparently that means maybe the bicycle normally costs $300, but you only paid $150, so you saved $150 due to the sale. If so then your savings was $150, not $135. It's like saying you're 150 years old not 135, even though having lived 150 years, you also lived 135 years. One includes the other, but they aren't the same. Not sure what this may have to do with geometry though.