Question

write each related conditional and determine each truth - value for the following conditional. see examples 4 and 5. if an angle measures 100, then it is obtuse. 26. converse 27. contrapositive 28. inverse 29. an employee at an animal shelter wrote the true conditional “if 47% of the dogs at the shelter are female, then 53% of the dogs are male.” can he rewrite this as a true biconditional? explain. see example 6. write two conditionals from each biconditional. see example 7. 30. a month has exactly 28 days if and only if it is february. 31. two angles are complementary if and only if their measures add up to 90. 32. the area of a square is s² if and only if the perimeter of the square is 4s.

Explanation:

Step1: Recall converse definition

The converse of a conditional "if p then q" is "if q then p". The original conditional is "If an angle measures 100, then it is obtuse". So the converse is "If an angle is obtuse, then it measures 100". An obtuse angle is an angle with measure greater than 90 and less than 180, so there are many obtuse - angled measures other than 100. The truth - value of the converse is false.

Step2: Recall contrapositive definition

The contrapositive of "if p then q" is "if not q then not p". For "If an angle measures 100, then it is obtuse", the contrapositive is "If an angle is not obtuse, then it does not measure 100". If an angle is not obtuse (acute or right - angled), its measure is not 100. The truth - value of the contrapositive is true.

Step3: Recall inverse definition

The inverse of "if p then q" is "if not p then not q". For "If an angle measures 100, then it is obtuse", the inverse is "If an angle does not measure 100, then it is not obtuse". There are non - 100 degree obtuse angles, so the truth - value of the inverse is false.

Step4: Analyze biconditional for dog problem

A biconditional "p if and only if q" is true when both "if p then q" and "if q then p" are true. For "If 47% of the dogs at the shelter are female, then 53% of the dogs are male", the converse is "If 53% of the dogs are male, then 47% of the dogs are female". Since the sum of the percentage of male and female dogs must be 100%, both the conditional and its converse are true. So it can be rewritten as a true biconditional.

Step5: Write conditionals from biconditional for month problem

For the biconditional "A month has exactly 28 days if and only if it is February", the two conditionals are: "If a month has exactly 28 days, then it is February" and "If it is February, then the month has exactly 28 days" (ignoring leap years for simplicity).

Step6: Write conditionals from biconditional for angle problem

For the biconditional "Two angles are complementary if and only if their measures add up to 90", the two conditionals are: "If two angles are complementary, then their measures add up to 90" and "If the measures of two angles add up to 90, then the two angles are complementary".

Step7: Write conditionals from biconditional for square problem

For the biconditional "The area of a square is s² if and only if the perimeter of the square is 4s", the two conditionals are: "If the area of a square is s², then the perimeter of the square is 4s" and "If the perimeter of the square is 4s, then the area of the square is s²".

Answer:

1. Converse: If an angle is obtuse, then it measures 100. False.
2. Contrapositive: If an angle is not obtuse, then it does not measure 100. True.
3. Inverse: If an angle does not measure 100, then it is not obtuse. False.
4. Yes, because both the conditional "If 47% of the dogs at the shelter are female, then 53% of the dogs are male" and its converse "If 53% of the dogs are male, then 47% of the dogs are female" are true.
5. Conditional 1: If a month has exactly 28 days, then it is February. Conditional 2: If it is February, then the month has exactly 28 days.
6. Conditional 1: If two angles are complementary, then their measures add up to 90. Conditional 2: If the measures of two angles add up to 90, then the two angles are complementary.
7. Conditional 1: If the area of a square is s², then the perimeter of the square is 4s. Conditional 2: If the perimeter of the square is 4s, then the area of the square is s².