PLANE AND SOLID GEOMETRY John Michael S. Mendoza, LPT, MA Divine Word College of Calapan Recall of Basic Geometric Terms Brief History The word "geometry" can be interpreted literally as “earth measure“, but this literal meaning seems far removed from the various modern geometries we are going to explore in this course. The idea of earth measure arises from the ancient pre~Greek development or geometry. These practical Egyptian and Babylonian applications of geometry involved measurement to a great extent, and they were not complicated by formal proofs. Brief History Brief History The explicit 3-4-5 example of Theorem, for instance, was used by determining a square corner for a field pyramid long before the theorem as established. the Pythagorean the Egyptians in or the base of a we know it was Brief History Euclid’s treatment of the subject has had an enormous influence on mathematics ever since, so much so that deductive reasoning is the method of mathematical inquiry today. In fact, this is often interpreted as meaning “geometry is 2-column proofs”. In other words, geometry is a formal axiomatic structure – typically the axioms of Euclidean plane geometry - and one objective of this course is to develop the axiomatic approach to various geometries, including plane geometry. This is a very important, though limited, interpretation of the need to study geometry, as there is more to learn from geometry than formal axiomatic structure. Brief History Evidence of geometry is found in all cultures. Geometric patterns have always been used to decorate buildings, utensils and weapons, reflecting the fact that geometry underlies the creation of design and structures. Patterns are visually appealing because they often contain some symmetry or sense of proportion. Symmetries are found throughout history, from dinosaur tracks to tire tracks. Buildings remain standing due to the rigidity of their triangular structures. Interest in the faithful representation of a three-dimensional scene as a flat two-dimensional picture has led artists to study perspective. Brief History In turn perspective drawing led to the introduction of projective geometry, a different geometry from the plane geometry of Euclid. The need for better navigation as trading distances increased along with an ever more sophisticated understanding of astronomy led to the study of spherical geometry. But it wasn’t until the 19th century, as a result of a study examining the role of Euclid’s parallel postulate, that geometry came to represent the study of the geometry of surfaces, whether flat or curved. Finally, in the 20th century this view of geometry turned out to be a vital component of Einstein’s theory of relativity. Thus, through practical, artistic and theoretical demands, geometry evolved from the flat geometry of Euclid describing one’s immediate neighborhood, to spherical geometry describing the world, and finally to the geometry needed for an understanding of the universe. Axioms and Postulates Today, the words axiom and postulate are used interchangeably. he development of geometry, however, the 'word postulate was used for an assumption confined to one particular subject (such as eometry), while axiom denoted a "universal truth," a more general assumption that applied to all of mathematics. Theorem Theorems are statements to be proved by using the axioms, definitions, and previous theorems as reasons for the logical steps in the proof. The theorems of geometry are valid conclusions based on the axioms. A simple theorem typically is stated in the form of an If~then statement. Theorem DEDUCTION THEOREM. If, under a given true premise p ,the conclusion q is always true; then the entire statement p implies q is true. FIRST POSTULATE ● A straight line may be drawn from any one point . This postulate tells that at least one straight line passes through two distinct points , but it does not say that there cannot be more than one such line . But Euclid generally assumed without mentioning that there is a unique line joining two points . SECOND POSTULATE ● A terminated line can be produced indefinitely. This postulate shows that what we call a line segment today was called a terminated line by Euclid .so the postulate says that a line segment can be extended on either side to make a line. THIRD POSTULATE A circle can be drawn with any centre and any radius. ● The meaning is very simple , this postulate shows that using an instrument a circle can be drawn at any plane because the radius remains the same. ● FOURTH POSTULATE All right angles are equal. ● This postulate shows that every right angle is equal and of 90° and right angles will remain same . So they will be equal . ● FIFTH POSTULATE ● If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less the two right angles , then the two straight lines , if produced indefinitely meet on which the sum of angles is less than two right angles The Fifth postulate is of a more technical nature, however. To understand what it is saying we need the notion of parallel lines.
