Geometry · Unit 1
Write a conjecture from a pattern, give a counterexample, and tell inductive from deductive.
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Inductive reasoning
Look at a pattern, then write a conjecture.
Example: 5, 10, 15, 20, … The next number is 25.
Counterexample
One example that makes a conjecture false is enough.
Example: Conjecture: All months have 31 days. Counterexample: February.
Deductive reasoning
Use facts to reach a conclusion that must be true.
Example: All even numbers are divisible by 2. 18 is even. Therefore 18 is divisible by 2.
Postulate, theorem, proof
Postulate: a statement accepted as true without proof.
Example: Through any three noncollinear points there is exactly one plane.
Theorem: a statement that must be proven.
Example: The sum of the angles in a triangle is 180°.
Proof: a valid justification that a statement is true.
Example: a chain of if-then statements that shows why the angles in a triangle add to 180°.
Find a pattern or a conclusion. Then say inductive or deductive, and why.
Each time is 25 minutes after the one before.
Next: 10:25 a.m., 10:50 a.m., 11:15 a.m.
The appointment times are every 25 minutes, starting at 9:10 a.m.
Inductive. You used a pattern of examples to write a conjecture, not a proof.
Answer: 10:25, 10:50, 11:15; inductive, from a pattern
Each term is 3 more than the one before.
Next: 16.
Each term is 3 more than the previous term, starting at 4.
Inductive. You used a pattern of examples to write a conjecture, not a proof.
Answer: 16; inductive, from a pattern
Each term is divided by 3.
Next: 1.
Each term is one-third of the previous term, starting at 81.
Inductive. You used a pattern of examples to write a conjecture, not a proof.
Answer: 1; inductive, from a pattern
The conclusion comes from a pattern of examples, not from a fact that must be true.
Inductive.
Answer: inductive, from a pattern of examples
The conclusion uses a definition and a given fact. If those are true, the conclusion must be true.
Deductive.
Answer: deductive, from a definition and a given fact
2 is prime and even.
Deductive. You used a specific fact that the conjecture cannot allow, so the conjecture is false.
Answer: 2; deductive, one fact disproves the conjecture
Name the type of reasoning and why. Write a conjecture, or give a counterexample.
15 ends in 5 and is not divisible by 10.
Deductive. You used a specific fact that the conjecture cannot allow, so the conjecture is false.
Answer: 15; deductive, one fact disproves the conjecture
Each time is 32 minutes later.
Next: 8:16 a.m., 8:48 a.m.
The bus comes every 32 minutes, starting at 6:40 a.m.
Inductive. You used a pattern of examples to write a conjecture, not a proof.
Answer: 8:16, 8:48; inductive, from a pattern
12 is divisible by 4 and not by 8.
Deductive. You used a specific fact that the conjecture cannot allow, so the conjecture is false.
Answer: 12; deductive, one fact disproves the conjecture
Two angles of 80° and 100° in different parts of a figure can be supplementary without being adjacent.
Deductive. You used a specific fact that the conjecture cannot allow, so the conjecture is false.
Answer: two supplementary angles that are not adjacent; deductive
The conclusion is a guess from examples, not a proof.
Inductive.
Answer: inductive, from examples
The conclusion uses a theorem and arithmetic. If the facts are true, the conclusion must be true.
Deductive.
Answer: deductive, from a theorem and given measures
Name the type of reasoning and why. Sort postulate, theorem, and proof.
A. Two points determine a line.
B. Vertical angles are congruent.
C. A sequence of if-then statements that shows why vertical angles are congruent.
What type of reasoning is this, and why?
A postulate is accepted as true: A.
A theorem must be proven: B.
A proof is a valid justification: C.
Deductive. You used the definitions of postulate, theorem, and proof.
Answer: A postulate, B theorem, C proof; deductive, from definitions
T(n) = 3n + 1
T(8) = 3(8) + 1
T(8) = 25
Deductive. You used a formula as a fact, so the value must be true.
Answer: 25; deductive, from a formula
Valid is about the chain, not about whether each sentence is true in the real world.
If all cats fly, and Miso is a cat, then Miso flies.
The chain is valid (detachment). The first sentence is false.
Yes.
Deductive. You used a law of logic (detachment) to check the chain.
Answer: Yes. Valid can use a false premise. Deductive, from a law of logic.
A non-square rhombus has four congruent sides and is not a square.
Deductive. You used a specific fact that the conjecture cannot allow, so the conjecture is false.
Answer: a non-square rhombus; deductive, one fact disproves the conjecture
These are triangular numbers.
Each term adds one more than the last addend: +2, +3, +4, +5, …
D(n) = n(n + 1)2
Inductive. You used a pattern of examples to write a conjecture, not a proof.
Answer: D(n) = n(n + 1)2; inductive, from a pattern
The first sentence is true.
The argument uses the converse, which is not a law of logic.
The conclusion need not follow. Invalid.
The conclusion may also be false (a non-square rectangle).
This is an invalid attempt at deductive reasoning. It does not follow a law of logic.
Answer: invalid (affirming the conclusion); not valid deductive reasoning
A. If two planes intersect, then they intersect in a line.
B. Linear pair angles are supplementary.
C. A sequence of if-then statements that shows why linear pair angles are supplementary.
What type of reasoning is this, and why?
A postulate is accepted as true: A.
A theorem must be proven: B.
A proof is a valid justification: C.
Deductive. You used the definitions of postulate, theorem, and proof.
Answer: A postulate, B theorem, C proof; deductive, from definitions
A. A chain of if-then statements that shows why right angles are congruent.
B. Right angles are congruent.
C. A plane contains at least three noncollinear points.
What type of reasoning is this, and why?
A proof is a valid justification: A.
A theorem must be proven: B.
A postulate is accepted as true: C.
Deductive. You used the definitions of postulate, theorem, and proof.
Answer: A proof, B theorem, C postulate; deductive, from definitions
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