Modern Geometry - Part 1

Front 1

Existence Axiom

Back 1

The collection of all points forms a nonempty set. There is more than one point in that set.

Front 2

Linear Pair Theorem:

If 2 angles form a linear pair, they are supplementary

Back 2

proof goes here

Front 3

Vertical angles Theorem:

Vertical Angles are Congruent

Back 3

proof goes here

Front 4

Angle-Side-Angle (ASA)

If ∆ABC & ∆DEF are 2 triangles such that ∠CAB ≅ ∠FDE, line segment AB ≅ life segment DE, & ∠ABC ≅ ∠DEF, then
∆ABC ≅ ∆DEF

Back 4

Let ∆ABC & ∆DEF be 2 triangles such that ∠CAB ≅ ∠FDE, line segment AB ≅ life segment DE, & ∠ABC ≅ ∠DEF, by hypothesis. We want to show that ∆ABC ≅ ∆DEF.

There exists a point C' on ray AC such that line segment AC' ≅ line segment DF by the Ruler Axiom.

Now, ∆ABC' ≅ ∆DEF (by SAS) and so, ∠ABC' ≅ ∠DEF by definition of congruent triangles.

Since ∠ABC ≅ ∠DEF by hypothesis, we conclude that ∠ABC ≅ ∠ABC'.

Then, by the Protractor Postulate,
ray BC = ray BC'

But ray BC' can only intersect line AC in at most one point, so C=C'