Chapter 1.7, Problem 5PSA

(a)

To determine

The Converse of given statements.

Answer to Problem 5PSA

1. If $B$ , then $A$ .
2. Wet $\Rightarrow$ Rain.
3. If an angle is acute, then it is $45°$ .
4. If a point divides a segment into two congruent segments, then it is the midpoint of the segment.

Explanation of Solution

Given Info:

The following statements are given

1. If A, then B
2. Rain $\Rightarrow$ Wet
3. If an angle is ${45}^o$ angle, then it is acute
4. If a point is the midpoint of the segment, it divides the segment into two congruent segments.

If a compound statement is given such that $A\Rightarrow B$ , then its converse statement is given by $B\Rightarrow A$ .

Therefore, the converse of the following statement would be

1. If $B$ , then $A$ .
2. Wet $\Rightarrow$ Rain.
3. If an angle is acute, then it is $45°$ .
4. If a point divides a segment into two congruent segments, then it is the midpoint of the segment.

(b)

To determine

The truth values of the converse statements given in part (a).

Answer to Problem 5PSA

1. If $B$ is true/false, then $A$ is true/false.
2. If statement: “wet” is true, then statement: “rain” may or may not be true.
3. If an angle is acute, then it is not necessary that it is $45°$ .
4. It is always true.

Explanation of Solution

Given Info:

The following statements are given

1. If A, then B
2. Rain $\Rightarrow$ Wet
3. If an angle is ${45}^o$ angle, then it is acute
4. If a point is the midpoint of the segment, it divides the segment into two congruent segments.

When we check the truth value of conditional statements, then we split it into two component statements and after that each component statement is checked for their truth values.

Therefore, the truth of the statements would be

1. If $B$ is true/false, then $A$ is true/false.
2. If statement: “wet” is true, then statement: “rain” may or may not be true.
3. If an angle is acute, then it is not necessary that it is $45°$ .
4. It is always true.