Chapter 4 Euclidean Geometry
4.1
Euclidean Parallelism, Existence of Rectangles
4.2
Parallelism and Trapezoids:
Parallel Projection
4.3
Similar Triangles, Pythagorean Theorem,
Trigonometry
4.5
The Circle Theorems
Homework 6
1

4.1
Euclidean Parallelism, Existence of Rectangles
We begin with a definition of parallel lines:
Two distinct lines L and M are said to be parallel iff they are in the same plane and do not
meet.
We also need the concept of a transversal.
A transversal is a line that intersects two other
distinct lines.
For example, the third side of a triangle is a transversal.
The author has a pictorial glossary for alternate interior angles, corresponding angles, and
interior angles on the same side of a transversal.
These terms apply to the angles whether
or not the lines are parallel.
4.1.1
Parallelism in Absolute Geometry
page 212
If two lines in the same plane are cut by a transversal so that a pair of alternate interior
angles are congruent, then the lines are parallel.
This theorem will be proved BEFORE we accept the axiom of parallel lines.
It shows
that parallel lines exist in Absolute Geometry.
Proof:
Let L and M be two lines cut by a transversal T.
Let angles 1 and 2 be alternate interior
angles.
Suppose L and M are not parallel and they intersect at some point R.
Then we have a triangle and one of the alternate interior angles is actually an exterior
angle of this triangle.
Since the other angle is interior, it is actually a remote interior
angle and by the Exterior Angle Inequality the exterior angle is strictly larger than the
remote interior angle.
This contradicts our hypothesis so our supposition is in error and
the lines are actually parallel.
Axiom P1
Euclidean Parallel Postulate
If L is any line and P any point not on L, then there exists in the plane of L and P one and
only one line that passes through P and is parallel to L.
With the scenario in the axiom, there are three possibilities for parallel lines:
•
none
(we’ve seen this with Spherical Geometry)
•
one
(this is the familiar Euclidean Geometry situation)
•
many (we’ll see this in Chapter 6 with Hyperbolic Geometry)
There are immediate and familiar theorems and corollaries that follow from making this
choice.
2

Theorem 4.1.1
page 214
The Transversal Theorems for Parallelism pages 216 & 217
Example 2
page 216 is the proof of
Theorem 4.1.3
Exterior Angle Theorem for
Euclidean Geometry.
Be sure to read it and enjoy the proof.
The corollary to 4.1.3 is one of the most famous
theorems in Euclidean Geometry.
It states that the sum of the interior angles of a triangle
is a constant 180.
Example 3
is the proof of yet another handy theorem
Theorem 4.1.4
The Midpoint Connector Theorem
page 218 and 219 is very well
done.
Please read the proof carefully.
It’s corollary is very handy indeed.