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

Explanation:

Step1: Recall the form of converse

The converse of "If p, then q" is "If q, then p". For the statement "If an angle measures 100, then it is obtuse", p is "an angle measures 100" and q is "it is obtuse".
Converse: If an angle is obtuse, then it measures 100.
This is false because an obtuse - angle is defined as an angle with measure greater than 90 and less than 180, not just 100.

Step2: Recall the form of contrapositive

The contrapositive of "If p, then q" is "If not q, then not p".
Contrapositive: If an angle is not obtuse, then it does not measure 100.
This is true. If an angle is not obtuse (i.e., its measure is less than or equal to 90 or greater than or equal to 180), then it cannot measure 100.

Step3: Recall the form of inverse

The inverse of "If p, then q" is "If not p, then not q".
Inverse: If an angle does not measure 100, then it is not obtuse.
This is false. For example, an angle could measure 120 (not 100) and still be obtuse.

Step4: Analyze biconditional for problem 29

A biconditional statement is of the form "p if and only if q" which means "If p, then q" and "If q, then p".
The original statement is "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 in a group of dogs is 100% (assuming only two genders), both the original conditional and its converse are true. So, it can be rewritten as a true biconditional.

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 original 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.