8.1 Points, Lines, Planes, and Angles 453 The History of Geometry Geometry as a science is said to have begun in the Nile Valley of ancient Egypt. The Egyptians used geometry to measure land and to build pyramids and other structures. The word geometry is derived from two Greek words: ge , meaning earth, and metron , meaning measure. Thus, geometry means “earth measure.” Unlike the Egyptians, the Greeks were interested in more than just the applied aspects of geometry. The Greeks attempted to apply their knowledge of logic to geometry. In about 600 b.c., Thales of Miletus was the first to be credited with using deductive methods to develop geometric concepts. Later, Pythagoras continued the systematic development of geometry that Thales had begun. In about 300 b.c., Euclid (see Profile in Mathematics ) collected and summarized much of the Greek mathematics of his time. In a set of 13 books called Elements , Euclid laid the foundation for plane geometry, which is also called Euclidean geometry . Euclid is credited with being the first mathematician to use the axiomatic method in developing a branch of mathematics. First, Euclid introduced undefined terms such as point, line, plane, and angle. He related these to physical space by such statements as “A line is length without breadth” so that we may intuitively understand them. Because such statements play no further role in his system, they constitute primitive or undefined terms. Second, Euclid introduced certain definitions . These definitions are introduced when needed and are often based on the undefined terms. Some terms that Euclid introduced and defined include triangle, right angle, and hypotenuse. Third, Euclid stated certain primitive propositions called postulates (now called axioms ) about the undefined terms and definitions. The reader is asked to accept these statements as true on the basis of their “obviousness” and their relationship with the physical world. For example, the Greeks accepted all right angles as being equal, which is Euclid’s fourth postulate. Fourth, Euclid proved, using deductive reasoning (see Section 1.1), other propositions called theorems . One theorem that Euclid proved is the Pythagorean theorem (see Section 8.3). He also proved that the sum of the angles of a triangle is 180°. Using only 10 axioms, Euclid deduced 465 theorems in plane and solid geometry, number theory, and Greek geometric algebra. Playing billiards involves many geometric concepts. In this section, we will introduce the geometric terms point, line, plane , and angle . In a typical billiard game, each of these terms is evident. A billiard ball rests on a specific point on the table, which is part of a plane. After being struck with a cue, the ball travels along a path that is part of a line. Angles are involved in the path the ball takes when the ball hits a table bumper or another ball. Points, Lines, Planes, and Angles SECTION 8.1 LEARNING GOALS Upon completion of this section, you will be able to: 7 Recognize and solve problems involving points and lines. 7 Recognize and solve problems involving planes. 7 Recognize and solve problems involving angles. Why This Is Important The concepts introduced in this section form a basis for the study of geometry. These concepts allow us to better apply geometry to solving a variety of problems that we encounter daily. These concepts are also used by scientists to describe the universe and by engineers to design and improve many of the products we rely on, including the digital camera on your smartphone. Profile in Mathematics Euclid Euclid (320–275 B.C. ) lived in Alexandria, Egypt, and was a teacher and scholar at Alexandria’s school called the Museum . Here Euclid collected and arranged many of the mathematical results known at the time. This collection of works became his 13-volume masterpiece known as Elements . Beginning with a list of definitions, postulates, and axioms, Euclid proved one theorem after another, using only previously proven results. This method of proof became a model of mathematical and scientific investigation that survives today. Remarkably, the geometry in Elements does not rely on making exact geometric measurements using a ruler or protractor. Rather, the work is developed using only an unmarked straightedge and a drawing compass. Next to the Bible, Euclid’s Elements may be the most translated, published, and studied of all the books produced in the Western world. FikMik/Shutterstock
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