Submitted by:Submitted to:
Jericah ManalangMrs. Joycelene Migano
Table of Contents
B. The Art of Reasoning
C. The Models of Points, Lines, and Angles
D. The Transversals
"Geometry," meaning "measuring the earth," is the branch of math that has to do with spatial relationships. In other words, geometry is a type of math used to measure things that are impossible to measure with devices. For example, no one has been able take a tape measure around the earth, yet we are pretty confident that the circumference of the planet at the equator is 40,075.036 kilometres (24,901.473 miles) . How do we know that? The first known case of calculating the distance around the earth was done by Eratosthenes around 240 BCE. What tools do you think current scientists might use to measure the size of planets? The answer is geometry. However, geometry is more than measuring the size of objects. If you were to ask someone who had taken geometry in high school what it is that s/he remembers, the answer would most likely be "proofs." (If you were to ask him/her what it is that s/he liked the least, the answer would probably be "proofs.") A study of Geometry does not have to include proofs. Proofs are not unique to Geometry. Proofs could have been done in Algebra or delayed until Calculus. The reason that High School Geometry almost always spends a lot of time with proofs is that the first great Geometry textbook, "The Elements," was written exclusively with proofs. This textbook is based on Euclidean (or elementary) geometry. "Euclidean" (or "elementary") refers to a book written over 2,000 years ago called "The Elements" by a man named Euclid. In the book, Euclid started with some basic concepts. He built upon those concepts to create more and more concepts. His structure and method influence the way that geometry is taught today. Euclid's book and interpretations of it were used as part of the curriculum of many high schools even until the beginning of the 20th century. Although this textbook is not a re-interpretation of The Elements, it will include more than just facts about geometric objects; the ability to "prove" that a particular answer is correct is part of the course.
The Art of Reasoning
Inductive reasoning is what we use most often. Inductive reasoning is reaching a conclusion based on previous observations. For example, if I notice that the sun rises in the east every day, then through inductive reasoning I could conclude that the sun will rise from the East tomorrow. In math, we may notice a pattern from which we draw conclusions.
Deductive reasoning is reaching a conclusion by combining known truths to create a new truth. Unlike inductive reasoning, deductive reasoning is certain, provided that the normal rules of logic are used to conclude such truths. In order to use deductive reasoning there must be a starting point, normally called the axioms or postulates of the theory. For example, an axiom in geometry asserts that given two points there is only one line that contains both points. Observe that while this is an axiom, it can be used to deduce that two different lines intersect in at most one point.
What you need to know
Conditional: a conditional is something which states that one statement implies another. A conditional contains two parts: the condition and the conclusion, where the former implies the latter. A conditional is always in the form "If statement 1, then statement 2." In most mathematical notation, a conditional is often written in the form p ⇒ q, which is read as "If p, then q" where p and q are statements. Converse:...