Elementary College Geometry

This plane geometry textbook is provided for free and open access through CUNY Academic Works and is part of their Open Educational Resources (OER). It's specifically hosted by the New York City College of Technology (NYCCT). The open access nature is emphasized, allowing for sharing, redistribution, and adaptation of the material under a Creative Commons license with attribution requirements. Contact information for CUNY Academic Works is provided for inquiries. The preface highlights the textbook's suitability for a 45-semester-hour introductory course and its role as a foundational text for more advanced mathematics courses like trigonometry, solid geometry, analytic geometry, and calculus. The emphasis is on practical application, using a clear and concise presentation style, easily understandable by students with a basic algebra background. The author, Henry Africk, is affiliated with CUNY's New York City College of Technology.

The textbook's approach prioritizes applying basic geometric principles to numerical problem-solving. To achieve this, the number of theorems and definitions is kept minimal, favoring short and intuitive proofs similar to those found in trigonometry or precalculus texts. The problems are structured in pairs, allowing for flexibility in assignments (odd or even-numbered problems). Numerous worked-out examples are provided to aid student understanding. Problems are presented primarily using diagrams to simplify the translation from words to visual representations. Many problems involve solving algebraic equations within geometric contexts, aiming to reinforce students' algebraic and numerical skills. A small number of exercises apply geometry to practical scenarios. The textbook aims to demonstrate the usefulness of geometry and enhance student appreciation through the inclusion of historical notes.

This text is explicitly intended for a brief introductory course in plane geometry, covering elementary geometry topics most relevant for subsequent advanced mathematics courses. The prerequisite is only a semester of algebra. The book is designed to be suitable for a course of approximately 45 semester hours. The focus remains on the practical application of fundamental geometric concepts to enhance numerical problem-solving skills. Historical notes are included to enrich students' understanding and appreciation of the subject matter. The overall goal is to provide a solid foundation in plane geometry that's both accessible and relevant for students continuing their mathematical education.

The section starts by establishing the foundational elements of plane geometry: points and lines. A line is defined as a straight line, and the unique relationship between two distinct points and the single straight line passing through them is highlighted. The notation for a line passing through points A and B is introduced as l-+ AB. The concept of the length of a line segment and its measurement (in inches, feet, meters, etc.) is introduced, emphasizing that the distance between two points is equivalent to the length of the line segment connecting them. An example problem is given, illustrating how to solve for an unknown length (x) within a line segment, although the text notes that units of measurement are often omitted for brevity. A historical note touches upon geometry's origins in practical problems, citing ancient civilizations' use of geometrical relationships in architecture, surveying, and astronomy, emphasizing that geometry originally served practical purposes before its theoretical development.

The concept of angles is introduced, defining an angle's vertex and sides with appropriate notation (∠BAC, ∠CAB, or simply ∠A). The text explains how to measure angles using a protractor, emphasizing the correct placement of the protractor on the vertex and the use of either the upper or lower scale for measurement. The section then shifts to the concept of postulates and axioms, stating that postulates are assumed true statements that do not require proof. It uses the assumption that angles can be measured in degrees as an example, referring to a prior example illustrating that fact implicitly. The importance of proof in mathematics is then mentioned; proofs are a reasoning process to establish new statements from known true statements. The text continues with descriptions of corresponding and alternate interior angles, visually represented by the letter 'Z'. The section then briefly mentions that if one pair of corresponding or alternate interior angles are equal, all other pairs are also equal, leading to a subsequent theorem (Theorem 2).

The definition of congruent triangles is provided: two triangles are congruent if one can be superimposed exactly over the other, resulting in the coincidence of corresponding sides and angles. The importance of identifying corresponding vertices from congruence statements is highlighted using examples (e.g., △ABC ≅ △DEF). The text then elaborates on how to sketch a triangle given specific angle and side lengths, illustrating this with an example using a protractor and ruler. The concept of congruence theorems is introduced. The text states that while two triangles are congruent if all corresponding sides and angles are equal, congruence can sometimes be concluded with partial information. The Side-Angle-Side (SAS) theorem is introduced, indicating that two triangles are congruent if two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle. A brief historical note connects the SAS theorem to Euclid's Elements.

This section introduces the concept of parallel lines and the angles formed when a transversal intersects them. Corresponding angles and alternate interior angles are defined and illustrated. The text uses visual aids to show how these angles form a 'Z' shape, even when the 'Z' is stretched or reversed. A key point is made: if just one pair of corresponding or alternate interior angles are equal, then all other pairs of corresponding and alternate interior angles are also equal. This observation leads directly to Theorem 2, stating the equivalence between equal corresponding angles and equal alternate interior angles. An example is provided, showing a diagram with parallel lines and a transversal, where corresponding angles are shown to be equal. The text further explains how this knowledge can be used to find all other angles in the diagram, highlighting the use of supplementary angles and vertical angles.

The section introduces the parallel postulate, stating that through a given point, only one straight line can be drawn parallel to a given line. The text then delves into the historical context of this postulate, noting that many prominent mathematicians unsuccessfully attempted to prove it as a theorem rather than accepting it as a postulate. The text emphasizes that mathematicians felt it wasn't as self-evident as a postulate should be, necessitating formal justification. The contribution of Nikolai Ivanovich Lobachevsky, a Russian mathematician, is mentioned. In 1826, Lobachevsky presented a geometry system based on the assumption that more than one line could be drawn through a point parallel to a given line. The section also links the parallel postulate to Euclid's Elements, highlighting that Euclid's decision to present it as a postulate rather than a theorem was indeed correct. The text notes that this fact wasn't widely understood until the 19th century, and that numerous unsuccessful attempts at proving the postulate were made in the intervening centuries. The section concludes by acknowledging that while the standard parallel postulate is most practical for everyday use, non-Euclidean geometries, such as Riemann's geometry, may provide better approximations in contexts involving vast distances, such as astronomy.

The section discusses methods for verifying the parallel postulate and its consequence—the angle sum theorem (that the sum of angles in a triangle is 180°). The text explains that one method is to test whether the sum of the angles in a triangle is indeed 180°. It acknowledges potential experimental errors in such tests and mentions that a small deviation from 180° doesn’t necessarily invalidate the theorem. However, the text highlights a significant challenge: Lobachevsky's non-Euclidean geometry proposes that the sum of angles in a triangle is always less than 180°. This discrepancy underscores the fundamental difference between Euclidean and non-Euclidean geometries and emphasizes that the validity of the parallel postulate is not universally accepted across all geometrical systems. The angle sum theorem (that the sum of a triangle's angles equals 180°) is also shown to be a critical consequence of the parallel postulate, providing another avenue to test the postulate's truthfulness.

The section introduces the Pythagorean Theorem, though the exact statement isn't explicitly given in this excerpt, it is implied to be a² + b² = c². The text immediately provides a historical perspective, highlighting that Pythagoras wasn't the first to discover it; evidence suggests that the Chinese, Babylonians, Egyptians, and Hindus knew of the theorem before his time. However, Pythagoras is credited with providing the first proof of the theorem, possibly using area-based methods. The text also mentions the existence of several hundred different proofs of the Pythagorean Theorem since Pythagoras' time, and that Pythagoras himself was the founder of the Pythagorean school, a secret society dedicated to the study of philosophy, mathematics, and science, implying a secretive nature to the early development and understanding of the theorem and its implications within the wider mathematical community.

While a specific proof isn't detailed, the text alludes to a possible method Pythagoras might have used, potentially involving areas. A diagram is mentioned (Figure 5) showing squares built on the sides of a right-angled triangle, suggesting an area-based approach to the proof of the theorem. The text notes that at least several hundred different proofs exist, showcasing the theorem's enduring significance and the multiple ways it can be mathematically demonstrated and explained. The excerpt doesn't delve into the details of any specific proof, but this implies the significance of the theorem, indicating it is central to geometry and warrants numerous approaches in demonstrating and explaining its validity.

The section begins by defining a polygon as a figure formed by line segments that enclose a portion of a plane. The line segments are called sides, the angles formed by the sides are the angles of the polygon, and the points where the sides meet are the vertices. The simplest polygon, the triangle (with 3 sides), is mentioned. The main focus of the section, however, is on quadrilaterals—polygons with 4 sides. The text also briefly lists other polygons, such as pentagons (5 sides), hexagons (6 sides), and octagons (8 sides), to give the reader a broader understanding of polygons beyond quadrilaterals and triangles. The description focuses on the basic structure and definition of polygons with a primary emphasis on quadrilaterals for further detailed study, indicating that the subsequent material will likely concentrate on the properties of four-sided figures.

While the provided text excerpt focuses on the introduction of polygons and specifically mentions quadrilaterals as the subject of the chapter, the detail provided is limited. The mention of other polygons (pentagons, hexagons, octagons) suggests that the section will likely continue exploring the general properties and characteristics of polygons beyond the quadrilateral. This implied continuation suggests that the text will further expand on the classification, properties, and possibly theorems related to different types of polygons, building upon the initial definition and categorization introduced in the first part of this section. The specific content of this implied continuation is not available within the provided text excerpt.

This plane geometry textbook is explicitly identified as a free and open-access resource, made available by the New York City College of Technology (NYCCT), which is part of the City University of New York (CUNY) system. The textbook's inclusion in CUNY Academic Works and its acceptance into the Open Educational Resources (OER) collection are clearly stated. This indicates a commitment from NYCCT and CUNY to provide accessible and high-quality educational materials to students. The open-access nature is further emphasized by the provided Creative Commons license terms, which allow for sharing, adaptation, and redistribution of the material with appropriate attribution. The contact information for CUNY Academic Works is also included for those seeking further information regarding the resource. The author, Henry Africk, is explicitly linked to CUNY and NYCCT.

A URL (https://academicworks.cuny.edu/ny_oers) is provided for users to find additional works related to this textbook. The license terms indicate that users are free to share and adapt the material, provided they give appropriate credit and indicate any changes made. The attribution requirement ensures that the original source (NYCCT and CUNY) is recognized for their work in creating and making this OER available. The restrictions on the license prevent any suggestion of endorsement by the licensor of the user's work. This detailed information is crucial for ensuring responsible use and promoting the open-access goals of CUNY and NYCCT in distributing this educational resource. The text also explicitly asks users to provide feedback on how access to this resource has benefited them.