![]() For example, the set of odd integers between 0 and 10 looks like this: A =. We usually name sets with capital letters. First, a set is a collection of objects joined by some common criteria. So, it seems appropriate to introduce a mini-version of set theory here. Union and intersection of sets is a topic from set theory that is often associated with points and lines. We also have C D ↔ ⊥ E F ↔ C D ↔ ⊥ E F ↔ the line containing the points C C and D D is perpendicular to the line containing the points E E and F F because both lines trace grid lines, which are perpendicular by definition.įinally, we see that C D ↔ ⊥ G H ↔ C D ↔ ⊥ G H ↔ the line containing the points C C and D D is perpendicular to the line containing the points G G and H H because both lines trace grid lines, which are perpendicular by definition. We can also state that A B ↔ ⊥ G H ↔ A B ↔ ⊥ G H ↔ the line containing the points A A and B B is perpendicular to the line containing the points G G and H H because both lines trace grid lines, which are perpendicular by definition. ![]() We know this because both lines trace grid lines, and intersecting grid lines are perpendicular. Therefore, we can safely say the following:Ī B ↔ ∥ C D ↔ A B ↔ ∥ C D ↔, the line containing the points A A and B B is parallel to the line containing the points C C and D D.Į F ↔ ∥ G H ↔ E F ↔ ∥ G H ↔, the line containing the points E E and F F is parallel to the line containing the points G G and H H.Ī B ↔ ⊥ E F ↔ A B ↔ ⊥ E F ↔, the line containing the points A A and B B is perpendicular to the line containing the points E E and F F. ![]() Additionally, all intersections form a 90 ∘ 90 ∘ angle. The grid also tells us that the vertical lines are parallel and the horizontal lines are parallel. Because they are on a grid, we assume all lines are equally spaced across the grid horizontally and vertically. The third way would be the most efficient way.ĭrawing these lines on a grid is the best way to distinguish which pairs of lines are parallel and which are perpendicular. From school to the post office or dry cleaners to home: first way A E ¯, E F ¯, F C ¯, C D ¯, D G ¯ A E ¯, E F ¯, F C ¯, C D ¯, D G ¯ second way A B ¯, B E ¯, E F ¯, F C ¯, C D ¯, D G ¯ A B ¯, B E ¯, E F ¯, F C ¯, C D ¯, D G ¯ third way A B ¯, B C ¯, C F ¯, F G ¯ A B ¯, B C ¯, C F ¯, F G ¯.The first way should be the most efficient way. From school to the library: first way A E ¯, E F ¯, F G ¯ A E ¯, E F ¯, F G ¯ second way A B ¯, B C ¯, C D ¯, D G ¯ A B ¯, B C ¯, C D ¯, D G ¯ third way A B ¯, B E ¯, E F ¯, F G ¯ A B ¯, B E ¯, E F ¯, F G ¯ fourth way A B ¯, B C ¯, C F ¯, F G ¯ A B ¯, B C ¯, C F ¯, F G ¯.It seems that the third way is the most efficient way. From school to the grocery store and home: first way A H ¯, H I ¯, I E ¯, E F ¯, F G ¯ A H ¯, H I ¯, I E ¯, E F ¯, F G ¯ second way A B ¯, B E ¯, E I ¯, I E ¯, E F ¯, F G ¯ A B ¯, B E ¯, E I ¯, I E ¯, E F ¯, F G ¯ third way A E ¯, E I ¯, I E ¯, E F ¯, F G ¯ A E ¯, E I ¯, I E ¯, E F ¯, F G ¯.Postulate 2: Any straight line segment can be extended indefinitely in a straight line.īefore we go further, we will define some of the symbols used in geometry in Figure 10.3:.Postulate 1: A straight line segment can be drawn joining any two points.He defined a point as “that which has no part.” It was later expanded to “an indivisible location which has no width, length, or breadth.” Here are the first two of the five postulates, as they are applicable to this first topic: The first definition Euclid wrote was that of a point. In order to write his postulates, Euclid had to describe the terms he needed and he called the descriptions “definitions.” Ultimately, we will work with theorems, which are statements that have been proved and can be proved. There were no formal geometric definitions before Euclid, and when terms could not be defined, they could be described. A postulate is another term for axiom, which is a statement that is accepted as truth without the need for proof or verification. In The Elements, Euclid summarized the geometric principles discovered earlier and created an axiomatic system, a system composed of postulates. These definitions form the foundation of the geometric theories that are applied in everyday life. In this section, we will begin our exploration of geometry by looking at the basic definitions as defined by Euclid. Determine union and intersection of sets.Express points and lines using proper notation.Identify and describe points, lines, and planes.(credit: modification of work “School of Athens” by Raphael (1483–1520), Vatican Museums/Wikimedia, Public Domain) Learning ObjectivesĪfter completing this section, you should be able to: Figure 10.2 The lower right-hand corner of The School of Athens depicts a figure representing Euclid illustrating to students how to use a compass on a small chalkboard.
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