TOC & Ch. 0 & Ch. 1 Axiom Table of Contents Ch. 0 Introduction Ch. 1 Axiomatic Systems 1.1.1 Introduction 1.1.2 Examples 1.1.3 History 1.2 A Finite Geometry 1.3 Finite Projective 1.4 Applications Ch. 2 Neutral Geometry Ch. 2 Table of Contents 2.1.1 Introduction 2.1.2 History 2.1.3 Analytic Models 2.2 Incidence Axioms 2.3 Distance/Ruler Axioms 2.4.1 Plane Separation Axiom 2.4.2 Angle & Measure 2.5.1 Supplement Postulate 2.5.2 SAS Postulate 2.6.1 Parallel Lines 2.6.2 Saccheri Quadrilateral 2.7.1 Euclid Parallel Postulate 2.7.2 Hyperbolic Parallel Postulate 2.7.3 Elliptic Parallel Postulate 2.8 Euclid/Hyperbolic/Elliptic Birkhoff's Axioms Hilbert's Axioms SMSG Axioms Ch. 3 Transformational Ch. 3 Table of Contents 3.1.1 Introduction 3.1.2 History 3.2.1 Definitions 3.2.2 Analytic Model 3.2.3 Affine Transformation 3.3.1 Isometry 3.3.2 Model/Collinearity 3.3.3 Model/Isometry 3.4.1 Direct Isometry 3.4.2 Model/Direct 3.5.1 Indirect Isometry 3.5.2 Model/Indirect 3.6.1 Similarity Transformation 3.6.2 Model/Similarity 3.7 Other Affine Transformations Ch. 4 Projective Geometry Ch. 4 Table of Contents 4.1.1 Introduction 4.1.2 Historical 4.2.1 Axioms 4.2.2 Basic Theorems 4.3 Duality 4.4 Desargue's Theorem 4.5.1 Harmonic Sets 4.5.2 Music & Harmonic Sets 4.6.1 Definitions for Projectivity 4.6.2 Fundamental Theorem 4.6.3 Projectivity/Harmonic Sets 4.6.4 Alternate Construction 4.7.1 Conics 4.7.2 Pascal's Theorem 4.7.3 Tangents to Conics Other Topics Ch. 5 Spherical Geometry Ch. 6 Fractal Geometry Ch. 7 Topology Appendices Internet Resources Index Geometer's Sketchpad/GeoGebra JavaSketchpad/GeoGebraHTML Video Lectures Logic Review References Acknowledgements

2.4.2 Angles and Angle Measure  Printout
The right angle from which to approach any problem is the try angle.

Definitions. An angle is the union of two noncollinear rays with a common endpoint. The common endpoint is called the vertex of the angle, and the rays are called the sides of the angle.

The Ruler Postulate and Ruler Placement Postulate were motivated by the "real-world" use of rulers. A similar set of postulates, SMSG Postulates 1113, which are motivated by the "real-world" protractor, do the same for angles. Hence, these axioms are sometimes referred to as the protractor postulates.

Postulate 11. (Angle Measurement Postulate) To every angle there corresponds a real number between 0 and 180.

Postulate 12. (Angle Construction Postulate) Let  be a ray on the edge of the half-plane H. For every r between 0 and 180, there is exactly one ray with P in H such that .

Postulate 13. (Angle Addition Postulate) If D is a point in the interior of , then .

Note that an angle has measure between 0 and 180. No angle has measure greater than or equal to 180, or less than or equal to 0.

Definitions.
Two angles are congruent if they have the same measure, denoted .
The interior of an angle  is the intersection of set of all points on the same side of line BC as A and the set of all points on the same side of line AB as C, denoted . (Note that this definition uses the Plane Separation Postulate.)

The interior of a triangle ABC is the intersection of the set of points on the same side of line BC as A, on the same side of line AC as B, and on the same side of line AB as C.
The bisector of an angle  is a ray BD where D is in the interior of  and .
A right angle is an angle that measures exactly 90.
An acute angle is an angle that measures between 0 and 90.
An obtuse angle is an angle that measures between 90 and 180.
Two lines are perpendicular if they contain a right angle.

The next theorem, stated here without proof, will be used in later sections.

Theorem 2.7.   and D is on the same side of line   as C if and only if .

Exercise 2.32.  Find the axioms from a high school book that correspond to SMSG Postulates 11, 12, and 13.

Exercise 2.33. Find the measures of the three angles determined by the points A(1, 1), B(1, 2) and C(2, 1) where the points are in the (a) Euclidean Plane; and  (b) Poincaré Half-plane.
Also, find the sum of the measures of the angles of the triangles.

Exercise 2.34. Find the angle bisector of , if A(0, 5), B(0, 3), and  where the points are in the (a) Euclidean Plane; and  (b) Poincaré Half-plane.

Exercise 2.35. Given , , and . Prove or disprove .

Exercise 2.36. Prove or disprove that all right angles are congruent.

Exercise 2.37. Prove or disprove that an angle has a unique bisector.

Exercise 2.38. (a)  Prove that given a line and a point on the line, there is a line perpendicular to the given line and point on the line.
(b)  Prove the existence of two lines perpendicular to each other.

Exercise 2.39. Prove Theorem 2.7.

Exercise 2.40. Prove congruence of angles is an equivalence relation on the set of all angles.

 © Copyright 2005, 2006 - Timothy Peil