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Study Notes 012:
By William E. Steinman:
Geometry 5 Theorems:
Basic angle theorems:
A theorem is a proposition that has been or can be proved on the basis of explicit assumptions. Once proved, it can be used to prove other statements. Some people call these theorems principles. We have five geometric theorems that are proved using the postulates from Study Notes 011.
Theorem 1: All right angles are congruent.
Theorem 2: All straight angles are congruent.
Congruent means the angles coinciding exactly when superimposed.
Recall that a right angle is an angle of 90o.
A straight angle is an angle of 180o.
Theorem 3: Complements of the same or of congruent angles are congruent.
Theorem 4: Supplements of the same or congruent angles are congruent.
Recall these from Study Notes 010.
Complementary angles are angles whose total angle is 900.
Supplementary angles are two angles whose total angle is 180o.
Theorem 5: Vertical angles are congruent.
Recall that Vertical angles are two nonadjacent angles formed by two intersecting lines.
c
a b
d
So, angles a and b are congruent, also angles c and d are congruent.
The logical concept of the hypothesis and the conclusion is encountered in math and in pretty much all of our life situations. It can takes the subject predicate form or the If, then form. In both cases, the hypothesis is given and conclusion is that which is to be proved.
For example, in the statement shaken beer foams, shaken beer is the hypothesis and foams is the conclusion. This is the subject predicate form. Harebrained kids prove this statement on a regular basis. What a waste of beer.
We could make the same statement in the “if, then” form.
If beer is shaken, then it foams. Same kids, same results.
The Converse of a statement is obtained by interchanging the hypothesis and conclusion.
For example, if it foams beer is shaken.
That brings us to the realization or rule that the converse of a true statement is not necessarily true.
There is more than one way to make beer foam.
However, our next rule is, the converse of a definition is always true.
For example, the definition of a square is a rectangle with four equal sides.
The converse is a rectangle with four equal sides is a square.
There are some obvious steps for proving a theorem.
First, we can divide the theorem into its hypothesis and its conclusion.
In plain language, we divide the statement into what is given and what is concluded.
In the above example, Shaken beer is given and foams is to be proved.
Just don’t do in with my beer.
This is a simple and obvious statement.
In cases where the proof is not simple and obvious other steps may be necessary to prove the theorem.
It may be necessary to make a diagram or sketch.
In complex geometric theorems, a drawing is often necessary.
It may also be necessary to make a plan of procedure.
This simply depends on the complexity of the theorem and the necessity of clearly demonstrating its validity.