Geometry · Unit 1
Draw a nested diagram for an if-then, plot people, and say what you can conclude.
If p, then q
Draw a small circle p completely inside a larger circle q. Everything in p is also in q.
Do not put an x yet. This picture is the rule for everyone, not one person or object.
Anyone outside q is automatically outside p.
Inside the inner circle
Think of p as a floaty inside of the pool q. If you are in the floaty, then you are in the pool.
Put an x only when the problem names a specific person or thing, or uses a word like “some.” The x stands for that one thing.
If x is in the floaty, then x is in the pool.
The contrapositive is the same picture
If you are not in the pool, then you are not in the floaty. That is ∼q → ∼p.
Example: If it is a puppy (p), then it is cute (q). Item x is not cute. Put x outside q. Then x is not a puppy.
Only in the outer ring
If you only know that you are in the swimming pool, you may or may not be in the floaty. No conclusion can be made.
Read a nested Euler diagram. Who is inside, in the ring, or outside?
Kai is inside the inner circle.
Kai may leave early.
Answer: inner circle; Kai may leave early
Riley is in “leave early,” but not necessarily in “lab done.”
No conclusion about whether Riley completed the lab.
Answer: no conclusion about the lab
Outside “leave early” is also outside “lab done.”
Jess did not complete the lab.
Answer: outside; Jess did not complete the lab
Outer region: rhombus. Inner region: square.
Answer: square inside rhombus
Outer: may park. Inner: has permit.
Answer: permit inside may park
Tessa could be in the outer ring only.
No.
Answer: no
Converse: switch. If a figure is a quadrilateral, then it is a kite.
The inner and outer regions swap.
Answer: quadrilateral inside kite
Inverse: If you do not have a library card, then you may not check out a book.
The inverse is the same picture as the converse: swap inner and outer.
Answer: may check out a book inside has library card
Draw the diagram for a conditional or for its contrapositive, then plot people.
Devon is in the outer region. That does not force the inner region.
No.
Answer: no
If you may not leave early, then you did not complete the lab.
That is the same nested picture: lab done inside leave early.
Answer: same diagram as the original conditional
ABCDE has 5 sides.
Answer: ABCDE has 5 sides
Q is not a pentagon.
Answer: Q is not a pentagon
No conclusion about orchestra.
Answer: no conclusion about orchestra
Omar is not in the inner circle.
He could be in the outer ring or outside both circles.
No conclusion about reading music.
Answer: no conclusion about reading music
Converse: If a student has a Saturday tournament, then the student is on the debate team.
The inner and outer regions swap.
Answer: Saturday tournament inside debate team
Inverse: If the oven is not on, then the kitchen is not warm.
The inverse is the same picture as the converse: swap inner and outer.
Answer: kitchen warm inside oven on
The contrapositive uses the same nesting. An X outside the outer region is ∼q.
If players warm up, then they run faster.
Contrapositive: If players do not run faster, then they did not warm up.
The X is outside “runs faster,” so it is also outside “warmed up.”
Answer: same nested diagram; X outside the outer region
Contrapositive: If a figure does not have four right angles, then it is not a square.
Answer: square inside four right angles
Asha did not buy a ticket.
Answer: Asha did not buy a ticket
Ben has a seat.
No conclusion about whether Cam bought a ticket.
Answer: Ben has a seat; no conclusion for Cam
40 is divisible by 10, so it is divisible by 5.
35 is divisible by 5, so it belongs in the outer region. It is not divisible by 10.
Answer: 40 is in the inner circle; 35 is in the outer ring only
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