Deduction in Geometry
Deductive geometry is a process of deriving geometric facts from previously-known facts by
using logical reasoning.
The Logical Form
We use letters like P and Q to represent simple statements.
Premise 1: If P, then Q.
Premise 2: P is true.
Conclusion: Therefore, Q must be true.
If P, then Q can also be read as "P implies Q"
The notation is given as .
To make this easier to remember, replace the letters P and Q with actual events:
Example 1:
Statement P: It is raining.
Statement Q: The grass is wet.
The argument:
1. If it is raining (P), then the grass is wet (Q).
2. It is raining (P).
Conclusion: Therefore, the grass is wet (Q).
This is a valid argument.
Example 2:
Statement P: Dev lives in Johor Bahru.
Statement Q: Dev lives in Malaysia.
The argument:
1. If Dev lives in Johor Bahru, then he lives in Malaysia.
2. Dev lives in Malaysia.
Conclusion: Dev lives in Johor Bahru.
Even though the statement is true, it does not enable us to draw a valid conclusion about P.
If Dev lives in Malaysia, he might live in Johor Bahru. He could also live in Kuala Lumpur, or any other places in Malaysia.
These possibilities are counterexamples to disprove the validity of the argument.
This is an invalid argument.
Deductive reasoning is about reaching conclusions that must be true.
Example 3:
Statement P: and are corresponding angles.
Statement Q: and are equal.
The argument:
1. If and are corresponding angles, then they are equal.
2. and are corresponding angles.
Conclusion: and are equal.
This is a valid argument.
Now you try to make an argument related to geometry.
In geometry, a logical argument is called proof, where you make your premises and conclusions.
