The Axioms of Equality

In the previous post, we have discussed about the axioms of real numbers. In this post, we learn about five more axioms, the axioms of equality in real numbers $latex \mathbb{R}$. Again, most of you are already familiar with these axioms since they are quite intuitive. We will emphasize these axioms because we are going to use them in proving theorems later.

Axiom E1: Relfexivity of Equality

For any a in $latex \mathbb{R}$, a = a.

Axiom E2: Symmetry for Equality

For any a and b in $latex \mathbb{R}$, if a = b, then b = a.

Axiom E3: Transitivity of Equality

For any a, b, and c in $latex \mathbb{R}$, if a = b and b = c, then a = c.

Axiom E4: Addition Property of Equality

For any a, b, and c in $latex \mathbb{R}$, if a = b, then a + c = b + c.

Axiom E5: Multiplication Property of Equality

For any a, b, and c in $latex \mathbb{R}$, if  a = b, then ac = bc.

 

The Exterior Angle Theorem

Introduction

If one side of a triangle is extended, an exterior angle is formed.  An exterior angle of a triangle forms a linear pair with the adjacent interior angle. The two non-adjacent interior angles to the exterior angle are called its remote interior angles. In the figure below, $latex \angle 1$ and $latex \angle 2$ are remote angles of $latex \angle 4$.

exterior angle

In the second triangle above, we can see that the sum of the measures of the remote angles is equal to the measure of the exterior angle. That is, $latex \angle 1 + \angle 2 = \angle 4$. Is this observation always true? In this post, we prove that it is indeed true Continue reading