Back
to Study Notes
Study Notes 010:
By William E. Steinman:
Geometry 3:
Adjacent angles are two angles, which have the same vertex (point of origin) and a common side.
B
W
Z
G
D E F
H
A Y Z
C
ABD and ACD are adjacent angles.
Vertical angles are two nonadjacent angles formed by two intersecting lines.
E and F are vertical angles. Also G and H are vertical angles.
Complementary angles are angles whose total angle is 900.
Supplementary angles are two angles whose total angle is 180o.
A
B C D
A and B are complementary angles.
C and D are supplementary angles.
Principles of pairs of angles:
Stating the obvious can get quite tedious.
1: If an angle X is cut into two adjacent angles Y and Z then Xo = Yo + Zo.
2: Vertical angles are congruent.
3: if two complementary angles contain Xo and Yo then Xo + Yo = 90o.
4: Adjacent angles are complementary if their exterior sides are perpendicular.
5: If two supplementary angles contain Yo and Zo then Yo + Zo = 180o.
6: Adjacent angles are supplementary if their exterior sides lie in the same straight line.
7: If supplementary angles are congruent, each of them is a right angle.
Deductive Reasoning or Deductive Logic:
In mathematics, propositions are established by a process of deductive reasoning. Also, in mathematics deductive reasoning, if done properly, is taken as acceptable proof.
The first person to discuss deduction was Aristotle, who proposed a number of argument forms called syllogisms.
Following Aristotle a school of philosophy known as Stoicism continued to develop deductive techniques of reasoning. Other contributors to these methods of proof were the German philosopher Gottlob Frege (1848 – 1925) and the German mathematician David Hilbert (1862 – 1943).
In deductive reasoning we have three steps.
First, we make a general statement, which refers to a whole class or set of things. This is a major premise.
Second, we make a particular or specific statement about one or more of the members of that set. This is a minor premise.
Third, we make a deduction, which follows logically when the general statement is applied to the specific statement. This is a conclusion.
These three steps form what Aristotle called a syllogism.
We might also point out that these steps and various perversions of them are a favorite game of simpering sophomores.
The process of deduction can be illustrated with a Venn diagram. This graphic method was developed by the British logician, John Venn (1834 – 1923).
Inclusive set, Consciousness.
Humans.
Rednecks.
An example is in order.
Major premise: All humans are conscious.
Minor premise: Rednecks are humans.
Conclusion: Rednecks are conscious?
Oh, oh!
What went wrong?
Observations, measurements and experiments.
None of these can be taken as proof.
We only need listen to eyewitness accounts of the same event, such as an auto accident, to conclude that observation cannot serve as proof.
Measurement cannot serve as proof because of the limits of our ability to measure. As Heisenberg so eloquently explained it, it is impossible to determine both the position and momentum of a subatomic particle (such as the electron) with arbitrarily high accuracy.
Experiments cannot serve as proof
because we are limited in the number of cases we can actually test. At best we can only achieve statistical accuracy. Tom Dewey
will testify to the fallacy of believing in statistical methods.