If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# CA Geometry: Deductive reasoning

1-3, deductive reasoning and congruent angles. Created by Sal Khan.

## Want to join the conversation?

• I'm not sure if Sal was correct on number 3. I didn't get how he explained it.
• Think about it like this: in both arguments, there is a statement, followed by another statement that corresponds or relates to the first statement somehow. Then there is the key word 'therefore', prior to the conclusion made that was based on the two statements. Deductive reasoning is a little tricky, but just remember this basic outline and you'll be fine: statement, statement, 'therefore', conclusion. I hope this helps!
• On question 3, statement II, Sal comes to the conclusion that it indeed is sound deductive reasoning. My thought process was, however, that 1/4 for instance cannot be written as a repeating decimal, yet it is rational. The fact that pi cannot be written as a repeating decimal does not necessarily make it irrational, and therefore I went by A. Where am I wrong?
• Valid deductive reasoning does not test whether the premises are True. Instead, it only states what would be the case IF the premises are true. IF the premises are True, then the conclusion must also be True, in a properly constructed deductive argument. This is known as being logically valid.

The issue of whether the premises are actually True is covered under testing whether the argument is logically sound. In order to be logically sound, the argument must be both logically valid and it must have premises that actually are True.

Sal misused the terminology at the end of the video, where he said "this is sound deductive reasoning". He should have stated "this is valid deductive reasoning".

Thus, if it is the case that all rational numbers can be written as a repeating decimal, then π cannot be rational because it cannot be written as a repeating decimal. This is an example of use of deductive reasoning. This is logically valid, but it is not logically sound.

Whether a number is a terminating or repeating decimal depends on the number base you use. We use base 10 numbers, under which ⅓ is a repeating decimal and ¼ is a terminating decimal. However, in base 9, one third is a terminating "decimal" and one fourth is a repeating "decimal" (we don't actually call them "decimals" in non-base 10 number systems).

However, 0 is rational but cannot be written as a repeating decimal in any number base. Therefore, the premise is false: not all rational numbers can be written as repeating decimals. Thus the argument is a logically unsound argument even though it is logically valid.

Therefore, you are correct: the argument was logically unsound. It was, however, logically valid and should have been included as an example of valid deductive reasoning (which just so happened to be unsound).

Note: By definition, a rational number is a number that can be expressed as the ratio of two integers.
• Can we change the statement "A number can be written as a repeating decimal if it is rational. Pi cannot be written as a repeating decimal. Therefore, pi is not rational.” to “A boy can fit into the red booth if he is 100 lbs. Jimmy cannot fit into the red booth. Therefore, Jimmy is not 100 lbs.”?
• Yes, you can logically make that assumption given the first two statements. Good job on figuring that out!
• could you show an example of inductive reasoning?
• On question 2, The angles 3 and 4 are obviously not 90 degrees, So can't you just decipher the answer from that observation alone?
(1 vote)
• You CANNOT go by observation alone because the diagram might not be drawn to scale. After all, what looks like parallel lines might actually be angled at 0.1° and thus intersect at some point.

Thus, You can only use what is clearly established. The information you use must either be given or logically proven to be true.

In Problem 2, ∠3 and ∠4 are congruent. However, if and only if the transversal (t) intersects ℓ and m at a right angle, then ∠3 and ∠4 are also supplementary. But, since we don't know the angle, we cannot say one way or the other.

But we know that ℓ and m MUST be parallel because ∠3 and ∠4 are congruent.

SO, you cannot rely on what the diagram looks like, but only upon what has been clearly proved or given.
• i had a question whether a statement can be both deductive and inductive reasoning?
(1 vote)
• I can't think of any statement to say yes to your answer, so I think that it is not possible for a statement to be both deductive and inductive reasoning.

Deductive reasoning is making a specific conclusion based a general statement.
(just an example) Most students in grade 7 love to read books. Ian is a 7th grade student. From this, we can derive that Ian loves to read books.

Inductive reasoning is generalizing an idea based on specific statements.
(just an example) Ian, Tanya, and Tiffany love to read books. They are in 7th grade. From this, we generalize that 7th grade students love to read books.