A conditional is in the form "if P, then Q". We replace P and Q with logical statements. The common notation is to write [tex]P \rightarrow Q[/tex] to indicate that P causes Q to happen (more or less). Think of it as a flow diagram. We go from P to end up at Q.
The converse has us swapping P and Q to get [tex]Q \to P[/tex]
In the "if, then" form we would have "If Q, then P".
With this in mind, we can answer parts a through d below.
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Part a.
P = two lines are parallel
Q = corresponding angles are equal
Original = [tex]P \to Q[/tex]
Converse = [tex]Q \to P[/tex]
Converse = "If Q, then P"
Converse = "If corresponding angles are equal, then two lines are parallel"
The converse is true because of the converse of corresponding angles theorem.
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Part b.
P = the sum of angle A and angle B is 110 degrees
Q = angle C is 70 degrees
Converse = If Q, then P
Converse = "If angle C is 70 degrees, then the sum of angle A and angle B is 110 degrees"
This is true. If C is 70 degrees, then 180-C = 180-70 = 110 is the remaining amount of degrees for the other two angles (A+B).
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Part c.
P = alternate interior angles k and a are not equal
Q = two lines cut by the transversal are not parallel
Converse = "If Q, then P"
Converse = "If two lines cut by the transversal are not parallel, then the alternate interior angles k and a are not equal"
This is also true. Let's assume the opposite is the case for now, so that would mean that the converse is assumed to be false. This means statement Q is true and that leads to P being false. However, alternate interior angles being congruent means that we have parallel lines (converse of alternate interior angle theorem). Therefore, there is no way we could have Q true and P false. Updating P to true fixes things.
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Part d.
P = Johan throws coins in the fountain
Q = he (Johan) loses his money
Converse = "If Q, then P"
Converse = "If Johan loses his money, then he throws coins in the fountain"
This is not always true. There are other ways for Johan to lose his money either by spending it, misplacing it, dropping it, having it stolen, etc. Throwing coins in the fountain is just one of many ways for him to lose his money. Therefore, the converse is false.
