Geometry · Unit 1
Write if-then form, then the converse, inverse, and contrapositive.
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If p, then q
A conditional is p → q. p is the hypothesis. q is the conclusion.
Logically equivalent
A conditional and its contrapositive are logically equivalent. The converse and the inverse are logically equivalent to each other, not to the conditional.
Writing statements
Example: If it is Tuesday, then I will take a test. Write the contrapositive.
First rewrite the statement and write p and q above the parts.
| p | q | |||
| If | it is Tuesday | , then | I will take a test | . |
Contrapositive: ∼q → ∼p
If I don’t take a test, then it is not Tuesday.
This takes a little more work, but you will always get it right!
Write a sentence as If p, then q. Then write one related statement.
Hypothesis: a figure is a square.
Conclusion: it has four congruent sides.
If a figure is a square, then it has four congruent sides.
Answer: If a figure is a square, then it has four congruent sides.
If two angles are vertical, then they are congruent.
Answer: If two angles are vertical, then they are congruent.
Converse: switch.
q → p
If a figure has four right angles, then it is a rectangle.
Answer: If a figure has four right angles, then it is a rectangle.
Inverse: negate both.
∼p → ∼q
If a figure is not a rectangle, then it does not have four right angles.
Answer: If a figure is not a rectangle, then it does not have four right angles.
Contrapositive: switch and negate.
∼q → ∼p
If a figure does not have four right angles, then it is not a rectangle.
Answer: If a figure does not have four right angles, then it is not a rectangle.
“q when p” means If p, then q.
If you have a library card, then you can check out a book.
Answer: If you have a library card, then you can check out a book.
Watch “when,” “if,” “only if,” and “after.” Write one related statement.
p: It is raining.
q: The track is closed.
If p, then q.
If it is raining, then the track is closed.
Converse: q → p
If the track is closed, then it is raining.
Answer: If the track is closed, then it is raining.
p: You may use the 3D printer.
q: You finished the safety quiz.
“p only if q” means If p, then q.
If you may use the 3D printer, then you finished the safety quiz.
Inverse: ∼p → ∼q
If you may not use the 3D printer, then you did not finish the safety quiz.
Answer: If you may not use the 3D printer, then you did not finish the safety quiz.
If students have rehearsal, then they stay after school.
Contrapositive: If students do not stay after school, then they do not have rehearsal.
Answer: If they do not stay after school, then they do not have rehearsal.
Inverse: If two lines are not perpendicular, then they do not form a right angle.
Two lines can form a right angle only if they are perpendicular, so this inverse is true for lines in a plane. The original is true, so the inverse matches the converse, which is also true here.
Yes. The inverse is true.
Answer: If two lines are not perpendicular, then they do not form a right angle. Yes.
“p only if q” means If p, then q.
If a number is divisible by 10, then it is divisible by 5.
Answer: If a number is divisible by 10, then it is divisible by 5.
Converse: If a polygon has 5 sides, then it is a pentagon.
That is the definition of pentagon, so yes.
Answer: If a polygon has 5 sides, then it is a pentagon. Yes.
If a figure is a square, then it is a rhombus.
Converse: If a figure is a rhombus, then it is a square.
Answer: If square, then rhombus. Converse: If rhombus, then square.
p: The marching band takes the field.
q: The crowd stands up.
If p, then q.
If the marching band takes the field, then the crowd stands up.
Inverse: ∼p → ∼q
If the marching band does not take the field, then the crowd does not stand up.
Answer: If the marching band does not take the field, then the crowd does not stand up.
p: The field is frozen.
q: Practice is canceled.
“q if p” means If p, then q.
If the field is frozen, then practice is canceled.
Converse: q → p
If practice is canceled, then the field is frozen.
Answer: If practice is canceled, then the field is frozen.
p: The bus arrives.
q: Students scan their IDs.
If p, then q.
If the bus arrives, then students scan their IDs.
Contrapositive: ∼q → ∼p
If students do not scan their IDs, then the bus does not arrive.
Answer: If students do not scan their IDs, then the bus does not arrive.
True conditional, false converse. The contrapositive keeps the conditional’s truth.
Converse: If a figure is a rhombus, then it is a square. False (a non-square rhombus).
Inverse: If a figure is not a square, then it is not a rhombus. False.
Contrapositive: If a figure is not a rhombus, then it is not a square. True.
Only the contrapositive must be true.
Answer: Only the contrapositive must be true.
Converse: If your concert is on Friday, then you are in band.
Other groups can also have a Friday concert.
No.
Answer: If your concert is on Friday, then you are in band. No.
Inverse: If you are not in band, then your concert is not on Friday.
The inverse matches the converse, which is not forced.
No.
Answer: If you are not in band, then your concert is not on Friday. No.
Contrapositive: If your concert is not on Friday, then you are not in band.
A conditional and its contrapositive are logically equivalent.
Yes.
Answer: If your concert is not on Friday, then you are not in band. Yes.
If a student may park on campus, then the student has a permit.
Contrapositive: If a student does not have a permit, then the student may not park on campus.
Answer: If a student does not have a permit, then the student may not park on campus.
That switches and negates both parts.
It is the contrapositive.
Answer: contrapositive
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