Collin Barrett- Harvey 9-1
Geo 2
Outline of 2.5
A Need for a Proof
In this chapter, we learn that making a conjecture, can lead to a theorem.
After examining the relationship between the number of regions and number of dots on the circle, can you make a conjecture of the amount of regions inside the circle if there were six dots on the circle.
Ok so a common conjecture would be that every time a dot was added to the circle, the number of regions is doubled. But this is not true. The actual number of regions made with six dots is actually 31 regions, not 32. In order for a conjecture to be proven true, it needs to be proven deductively with a mathematical proof. To test your conjecture for this problem, you would have to draw a circle, mark six dots on the circle, and make the regions. After doing this, you will find that in fact 31 regions are made, not 32.
Inductive and Deductive Reasoning
Inductive reasoning is the process of forming conjectures based on basic observations. Since conjectures can be false, inductive reasoning is not accepted in mathematical proofs. Basically, proofs need to be supported by mathematical theorems/ postulates, or a drawing that proves your conjecture to be true or false.
(When you made a conjecture about the numbers of regions made by six dots on a circle, you proved your conjecture to be true or false by actually drawing the circle out.)
The following theorem is proven deductively, by evidence, below in a formal two-column proof:
Here are some additional theorems:
The following theorem is also proven deductively by a two column proof. (the answers are under the image)
25. Transitive Property or Substitution Property
26. Given
27. Subtraction Property
The Importance of Theorems
When forming a conjecture you should always think about a counterexample(s). Theorems differ from conjectures in that no matter the conditions, a theorem will always be true. For example, the Vertical Angles Theorem will always be true mo matter what the measure of the angles are.