The Timpson Lyceum · Geometry (2024)

Parallel Lines and Non-Euclidean Geometries – Lecture 10 (Geometry)

Overview

This lecture introduces the Parallel Postulate — Euclid's fifth and final postulate — and explores what happens to geometry when it is changed. We first define parallel lines and what it means for two rays to point in the same direction, then prove that through a point not on a line there exists at least one parallel line. The postulate asserts that this parallel is unique, and the lecture explains why uniqueness cannot be proved from the other axioms but must be assumed. Denying or weakening this single postulate produces the non-Euclidean geometries — hyperbolic and elliptic — which we survey along with their properties and a little of their history. The lecture closes with a chain of theorems that the postulate unlocks: transitivity of parallelism, the transversal angle relationships, the triangle angle sum, and the interior/exterior angle sums of convex polygons.

Key Concepts

Parallel lines — Two lines $l$ and $m$ are parallel, written $l \parallel m$ , if and only if they do not intersect. (Compare with perpendicularity, $l \perp m$ .)

Same-direction rays — Two rays have the same direction if and only if the lines through them are parallel and the rays lie on the same side of the line through their two endpoints (vertices).

Existence of a parallel (Theorem 8.1) — If $A$ is a point not on line $l$ , then there exists a line through $A$ parallel to $l$ . This is proved from earlier results (angle construction + the exterior angle inequality), so existence is never assumed.

The Parallel Postulate (Euclid's Fifth, Postulate 6.1) — If $A$ is a point not on line $l$ , then there exists one and only one line $m$ through $A$ with $m \parallel l$ . The uniqueness — not the existence — is the postulate. Roughly 2000 years of attempts to derive it from the other axioms failed, which is itself the proof that it must be taken as an independent assumption.

Non-Euclidean geometries — Replacing the uniqueness clause yields consistent alternative geometries:

Euclidean — a flat surface; exactly one parallel through the external point; triangle angle sum is exactly $180^\circ$ .
Hyperbolic (Gauss, also Bolyai and Lobachevsky) — a surface of negative, saddle-like curvature; infinitely many parallels through the external point; triangle angle sum is less than $180^\circ$ . Parallel lines do not stay equidistant — the gap grows without bound.
Elliptic (a Riemannian geometry, after Riemann) — a positively curved surface like a sphere; no parallels — any two lines meet (in two points); triangle angle sum is greater than $180^\circ$ .

Lines on a sphere (geodesics) — On the sphere a "line" is a great circle (e.g. the equator and the lines of longitude) — the shortest path between two points. Non-equatorial latitude circles are not lines; they are curves, since they do not give the shortest distance. This is why long flights trace curved arcs on a flat map.

Equidistance is not equivalent to the postulate — "Parallel lines stay the same distance apart" is true in Euclidean geometry but is just a logical equivalent of the postulate, not a proof of it; it fails in hyperbolic geometry.

Why a postulate, not a fact — Geometry only ever claims if the postulates hold, then the theorems hold. We don't assert which geometry is "true"; we ask which one models a given space. For our local space, inductively checking the axioms confirms Euclidean geometry, and then every Euclidean theorem (Pythagoras, the $180^\circ$ angle sum, etc.) follows for free.

Consequences of the postulate:

Transitivity of parallelism (Theorem): If $l \parallel m$ and $m \parallel n$ (distinct lines), then $l \parallel n$ . Parallelism is symmetric and transitive but not reflexive (a line is not parallel to itself), so it is not an equivalence relation.
Intersection theorem: If $l \parallel m$ and a line $n$ meets $l$ , then $n$ also meets $m$ .

Transversal terminology — A transversal is a line cutting two others, creating eight angles. The four between the two lines are interior, the four outside are exterior. Across the two intersections we get vertical angles, corresponding angles, alternate interior angles, and alternate exterior angles.

Transversal angle theorem (Theorem 8.4) — $l_1 \parallel l_2$ if and only if their alternate interior angles are congruent. The same statement holds (corollaries) for corresponding angles and for alternate exterior angles, by essentially the same proof.

Triangle angle sum (Theorem 8.5) — The interior angles of any triangle sum to a straight angle, $180^\circ$ .

Polygon angle sums (convex polygon with $n$ sides; one straight angle $= 180^\circ$ ):

Simple polygon — a closed polygonal path whose edges meet only at the (distinct) vertices.
Convex polygon — a simple polygon where the segment between any two of its points never crosses a side.
Regular polygon — both equilateral (equal sides) and equiangular (equal angles); neither condition implies the other (e.g. a rhombus vs. a rectangle).
Interior angle sum: $(n-2)\cdot 180^\circ$ — split the polygon into $n-2$ triangles.
Exterior angle sum: always $360^\circ$ (two straight angles), for every $n$ .

Worked Example

Exterior angle sum of a convex $n$ -gon. At each vertex, an interior angle and its exterior angle together form a straight angle, so the two sums combined over all $n$ vertices equal $n$ straight angles. Subtracting the interior-angle sum leaves the exterior-angle sum:

\begin{aligned} \text{Ext sum} &= n\cdot 180^\circ - \text{Int sum} \\ &= n\cdot 180^\circ - (n-2)\cdot 180^\circ \\ &= \bigl(n - n + 2\bigr)\cdot 180^\circ \\ &= 2\cdot 180^\circ = 360^\circ \end{aligned}

The $n$ cancels, so the exterior angles of any convex polygon sum to $360^\circ$ regardless of the number of sides.

Summary

$l \parallel m$ means the lines never intersect; same-direction rays must be parallel and on the same side of the line through their endpoints.
Existence of a parallel through an external point is a theorem; uniqueness is the independent Parallel Postulate (Euclid's Fifth) and cannot be derived from the other axioms.
Changing the uniqueness clause gives hyperbolic geometry (infinitely many parallels, saddle curvature, angle sum $< 180^\circ$ ) and elliptic geometry (no parallels, spherical curvature, angle sum $> 180^\circ$ ); on a sphere, "lines" are great circles.
Geometry asserts if the postulates hold, the theorems hold — we choose the geometry that best models a space rather than declaring one "true."
The postulate makes parallelism transitive and symmetric (but not reflexive) and proves the transversal angle relations: parallel lines $\iff$ congruent alternate interior (also corresponding, also alternate exterior) angles.
Triangle angle sum is exactly $180^\circ$ ; a convex $n$ -gon has interior-angle sum $(n-2)\cdot 180^\circ$ and a constant exterior-angle sum of $360^\circ$ .