Introduction

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:

Consider any triangle ABC in which the angles are aº, bº and cº.  Draw a line through A parallel to BC.

 

∴  PAB = b⁰                                                       {Alternate angles}

 QAC = c⁰                                                      {Alternate angles}

Now,  PAB +   BAC +   QAC = 180⁰      {Angle sum of a straight line}

∴ b⁰ + a_­­⁰ + c⁰ = 180⁰     {∴  PAB = b⁰ and   QAC = c⁰, proved above}

∴ a_­­⁰ + b⁰ + c⁰ = 180⁰

Hence the angle sum of a triangle is 180⁰.

Further theorems can now be deduced by using this theorem together with the axioms.  This is how the body of knowledge is increased using the deductive method.

The Exterior Angle of a Triangle

Theorem 2

Prove that the exterior angle of a triangle is equal to the sum of the interior opposite angles.

Proof:

Consider any triangle ABC in which the angles are aº, bº and cº.  Extend the line BC to the point D.

Read more below-

SS1 Mathematics Third Term: Deductive Proof