Question

for exercises 16 - 19, write the converse, inverse, and contrapositive of each conditional statement. determine the truth value of each statement. if it is false, provide a counterexample.

1. if a figure is a circle with radius r, then its circumference is 2πr.
2. if an integer ends with 0, then it is divisible by 2.
3. if you win the league championship game, then you win the league trophy.
4. if a triangle has one right angle, then the other two angles are complementary.
5. write the two conditionals that make up this biconditional: an angle is an acute angle if and only if its measure is between 0 and 90.

Explanation:

16.

Step1: Define converse

Switch hypothesis and conclusion.
Converse: If the circumference of a figure is $2\pi r$, then the figure is a circle with radius $r$. (False, counter - example: a non - circular closed curve with the same perimeter as a circle of radius $r$)

Step2: Define inverse

Negate both hypothesis and conclusion.
Inverse: If a figure is not a circle with radius $r$, then its circumference is not $2\pi r$. (False, counter - example: a non - circular closed curve with perimeter $2\pi r$)

Step3: Define contrapositive

Negate and switch hypothesis and conclusion.
Contrapositive: If the circumference of a figure is not $2\pi r$, then the figure is not a circle with radius $r$. (True)

Step1: Define converse

Switch hypothesis and conclusion.
Converse: If an integer is divisible by 2, then it ends with 0. (False, counter - example: 2, 4, 6, 8 etc.)

Step2: Define inverse

Negate both hypothesis and conclusion.
Inverse: If an integer does not end with 0, then it is not divisible by 2. (False, counter - example: 2, 4, 6, 8 etc.)

Step3: Define contrapositive

Negate and switch hypothesis and conclusion.
Contrapositive: If an integer is not divisible by 2, then it does not end with 0. (True)

Step1: Define converse

Switch hypothesis and conclusion.
Converse: If you win the league trophy, then you win the league championship game. (False, counter - example: could win trophy through other means like a points system and not win the final game)

Step2: Define inverse

Negate both hypothesis and conclusion.
Inverse: If you do not win the league championship game, then you do not win the league trophy. (False, counter - example: could win trophy through other means like a points system and not win the final game)

Step3: Define contrapositive

Negate and switch hypothesis and conclusion.
Contrapositive: If you do not win the league trophy, then you do not win the league championship game. (True)

Answer:

Converse: False (counter - example: non - circular closed curve with perimeter $2\pi r$); Inverse: False (counter - example: non - circular closed curve with perimeter $2\pi r$); Contrapositive: True

17.