Geometry · Unit 1
Put if-then statements in logical order, and prove a claim by contradiction.
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Rewrite as if-then
“When p, q,” “After p, q,” and “q if p” all become If p, then q. Then line the arrows up.
Logical order
A direct proof is a valid chain of if-then statements (syllogism). The conclusion of one line is the hypothesis of the next.
Indirect proof
Assume the opposite of what you want. Use facts until you reach a contradiction. Then the assumption is false, so the original claim is true.
Drop unused sentences
A list may include an extra sentence that does not fit the chain. Leave it out of the ordered proof.
Rewrite one sentence as if-then, then put a short chain in order.
“When p, q” means If p, then q.
If Priya gets the lead, then she memorizes her lines.
Answer: If Priya gets the lead, then she memorizes her lines.
“q if p” means If p, then q.
If Sam is eligible for the team, then he calls his coach.
Answer: If Sam is eligible for the team, then he calls his coach.
If Sam studies extra, then he earns a 90.
Answer: If Sam studies extra, then he earns a 90.
The two arrows already chain.
If the oven is hot, then the kitchen smells sweet.
Answer: If the oven is hot, then the kitchen smells sweet.
The conclusion of the first line must be “Nia boards the bus.”
If Nia leaves on time, then she boards the bus.
Answer: If Nia leaves on time, then she boards the bus.
If the paper is damp, then the printer jams.
Answer: If the paper is damp, then the printer jams.
Put a longer list in order so each conclusion feeds the next hypothesis.
If the review session is offered, then Sam studies extra.
If Sam studies extra, then he earns a 90.
If Sam earns a 90, then he is eligible for the team.
If Sam is eligible for the team, then he calls his coach.
Conclusion: If the review session is offered, then Sam calls his coach.
Answer: review → studies → 90 → eligible → calls coach
If the cable is plugged in, then the power is on.
If the power is on, then the router lights up.
If the router lights up, then the laptop connects.
Answer: cable → power → lights → connects
Drop: If the window is open, then the room is cool.
If the soil is damp, then the seeds are watered.
If the seeds are watered, then they sprout.
If they sprout, then the tray turns green.
Answer: soil damp → watered → sprout → tray green
If Priya gets the lead, then she memorizes her lines.
If she memorizes her lines, then she celebrates.
If she celebrates, then she calls her brother.
Answer: lead → lines → celebrates → calls brother
Law of Syllogism on the whole chain.
If Priya gets the lead, then she calls her brother.
Answer: If Priya gets the lead, then she calls her brother.
If the water boils, then the kettle whistles.
If the kettle whistles, then we pour tea.
The mugs sentence does not chain.
Chain the first two: If the water boils, then we pour tea.
Answer: If the water boils, then we pour tea.
Indirect proof: assume the opposite, reach a contradiction.
Assume the opposite: the collection does not contain a nickel.
Then the collection is only dimes.
The total of only dimes is a multiple of 10.
47 is not a multiple of 10.
That contradicts the given total.
The assumption is false, so the collection contains a nickel.
Answer: must contain a nickel
Assume there is no penny.
Then the collection is only quarters.
The total is a multiple of 25.
62 is not a multiple of 25.
Contradiction. So there is a penny.
Answer: must contain a penny
The angle sum is 180°.
40 + 70 + x = 180, so x = 70.
Assume it is right. Then one angle is 90°.
None of 40, 70, 70 is 90.
Contradiction. It is not a right triangle.
Answer: not a right triangle
Assume the opposite: n is even.
Then n = 2k for some integer k.
n² = 4k², which is even.
That contradicts n² odd.
So n is odd.
Answer: n is odd
Assume all three digits are odd.
The sum of three odd numbers is odd.
14 is even.
Odd cannot equal even.
Contradiction. So the three digits cannot all be odd.
Answer: cannot all be odd
Assume a triangle has two right angles.
Then two angles are 90° and 90°.
90 + 90 = 180, before adding the third angle.
The third angle would make the sum greater than 180°.
That contradicts the triangle angle sum of 180°.
So a triangle cannot have two right angles.
Answer: a triangle cannot have two right angles
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