| In deductive geometry, we do not accept any
other geometrical statement as being true unless it can be proved (or
deduced) from the axioms.
A statement that is proved by a sequence of logical steps is called a theorem.
To prove a theorem we start by using one or more of the axioms in a
particular situation to get some true statements. We then have to
apply logical reasoning to these statements to produce new statements that
are true. The proof ends when we arrive at the statement of the
theorem.
Properties of Equality
The relation of equality has the following properties. We will use
these properties in the proofs of some theorems.
1. Transitive Property
If a = b and b = c, then a = c.
2. Substitution Property
If a statement about a is true and a = b, then the
statement formed by replacing a with b (throughout) is also
true.
E.g. If the statement a + c = 180 is true and a is
equal to b, then the statement b + c = 180 is also
true.
Deductive Proofs of Theorems
To prove a theorem, draw a diagram. Write related statements and
give the reasons for each (i.e. state the axioms used). Then use the
transitive property and/or one of the other properties of equality.
Angle Sum of a Triangle
Theorem 1
Prove that the angle sum of a triangle is 180º.
Proof:
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