Chapter 4: Neutral Geometry §3.7 the Parallel Postulates and Models §4.1 the Exterior Angle Theorem and Existence of Perpendiculars

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Chapter 4: Neutral geometry §3.7 The Parallel Postulates and Models §4.1 The Exterior Angle Theorem and Existence of Perpendiculars

MTH 411/511

Foundations of Geometry

MTH 411/511 (Geometry) Euclid’s Elements Fall 2020 It’s good to have goals

Goals for today: • Introduce neutral geometry. • Deﬁne exterior angles and prove the Exterior Angles Theorem.

MTH 411/511 (Geometry) Euclid’s Elements Fall 2020 The parallel postulates and models

Let’s recall our three parallel postulates. Euclidean Parallel Postulate For every line ` and for every point P that does not lie on `, there is exactly one line m such that P lies on m and m k `.

Elliptic Parallel Postulate For every line ` and for every point P that does not lie on `, there is no line m such that P lies on m and m k `.

Hyperbolic Parallel Postulate For every line ` and for every point P that does not lie on `, there are at least two lines m and n such that P lies on both m and n and both m and n are parallel to `.

In neutral geometry (essentially what we have deﬁned so far), we will prove that parallel lines exist, so the Elliptic Parallel Postulate is inconsistent with the axioms of neutral geometry. However, we will show that both the Euclidean and the Hyperbolic Parallel Postulates are consistent.

MTH 411/511 (Geometry) Euclid’s Elements Fall 2020 The parallel postulates and models

Which models are consistent with our axioms? • ﬁnite geometries (fail the Ruler Postulate) 2 • the sphere S (fails the Incidence and Ruler Postulates) • the rational plane (fails the Ruler Postulate) 2 • the Cartesian plane R with the taxicab metric (fails SAS) 2 • the Cartesian plane R with the Euclidean metric (model with the Euclidean Parallel Postulate) • the Klein disk (model with the Hyperbolic Parallel Postulate) (what’s the metric???) In the next chapter we will begin to develop neutral geometry. In later chapters we 2 develop Euclidean Geometry (R + Euclidean metric) and Hyperbolic Geometry (Klein disk + some metric).

MTH 411/511 (Geometry) Euclid’s Elements Fall 2020 Neutral Geometry

Neutral geometry is the geometry (axiomatic system) with ﬁve undeﬁned terms: point, line, distance, half-plane, and angle measure, together with the following axioms: • The Existence Postulate • The Incidence Postulate • The Ruler Postulate • The Plane Separation Postulate • The Protractor Postulate • The Side-Angle-Side Postulate We will not assume any model and we will remain neutral on the parallel postulates (Euclidean vs Hyperbolic). Our goal in this chapter will be to prove the propositions in Book I of Euclid’s Elements (those that do not rely on the Euclidean Parallel Postulate.)

MTH 411/511 (Geometry) Euclid’s Elements Fall 2020 Exterior angles

Deﬁnition 1 Let 4ABC be a triangle. The angles ∠CAB, ∠ABC, and ∠BCA are called interior angles of the triangle. An angle that forms a linear pair with one of the interior angles is called an exterior angle for the triangle. If the exterior angle forms a linear pair with the interior angle at one vertex, then the interior angles at the other vertices are referred to as remote interior angles.

D B C E

In the diagram, angles ∠DCA and ∠ECB are exterior angles for the interior angle ∠BCA. The remote interior angles for these exterior angles are ∠ABC and ∠BAC. Note that ∠DCA and ∠ECB form a vertical pair.

MTH 411/511 (Geometry) Euclid’s Elements Fall 2020 Exterior angles

Exterior Angle Theorem (Theorem 4.1.2) The measure of an exterior angle for a triangle is strictly greater than the measure of either remote interior angle.

We’ll restate this so it’s easier to set up our proof.

D B C E

Exterior Angle Theorem (Theorem 4.1.2) – Restatement −→ −→ If 4ABC is a triangle and D is a point such that CD is opposite to CB, then µ(∠DCA) > µ(∠BAC) and µ(∠DCA) > µ(∠ABC).

MTH 411/511 (Geometry) Euclid’s Elements Fall 2020 Exterior angles

Exterior Angle Theorem (Theorem 4.1.2) −→ −→ If 4ABC is a triangle and D is a point such that CD is opposite to CB, then µ(∠DCA) > µ(∠BAC) and µ(∠DCA) > µ(∠ABC).

Proof. −→ −→ Let 4ABC be a triangle and D a point such that CD is opposite to CB. We will prove that that µ(∠DCA) > µ(∠BAC). The proof that µ(∠DCA) > µ(∠ABC) is similar. Let E be the midpoint of AC (Existence of Midpoints) and choose F to be the point −→ ∼ ∼ on BE such that BE = EF (PCP). Then ∠BEA = ∠FEC (Vertical Angles Theorem). ∼ ∼ Hence, 4BEA = 4FEC (SAS) and so ∠FCA = ∠BAC (deﬁnition of congruent 4s).

MTH 411/511 (Geometry) Euclid’s Elements Fall 2020 Exterior angles

Exterior Angle Theorem (Theorem 4.1.2) −→ −→ If 4ABC is a triangle and D is a point such that CD is opposite to CB, then µ(∠DCA) > µ(∠BAC) and µ(∠DCA) > µ(∠ABC).

Proof. Next we claim that F is in the interior of DCA. Note that F and B are on opposite ←→ ∠ ←→ sides of AC and B and D are on opposite sides of AC, so F and D are on the same ←→ ←→ side of AC (PSP). Also, A and E are on the same side of CD, and E and F are on the ←→ ←→ same side CD (The Ray Theorem). Thus, A and F are on the same side of CD (PSP). It follows that F is in the interior of ∠DCA as claimed (deﬁnition of angle interior). ∼ Now µ(∠DCA) > µ(∠FCA) (Btw Thm for Rays). Since ∠FCA = ∠BAC (above), then µ(∠DCA) > µ(∠BAC) as claimed.

2 Note that this theorem fails on the sphere S .

MTH 411/511 (Geometry) Euclid’s Elements Fall 2020 Next time

Before next class: Finish reading Section 4.1. Start reading Section 4.2.

In the next lecture we will: • Review deﬁnition of perpendicular and uniqueness of perpendiculars through a point on a line. • Prove existence and uniqueness of perpendiculars (in general). • Discuss various triangle congruence conditions

MTH 411/511 (Geometry) Euclid’s Elements Fall 2020