Points, Lines, and Planes

Conditional Statements, Inductive \u0026 Deductive Reasoning (Complete Geometry Course Lesson 2)
Conditional Statements, Inductive \u0026 Deductive Reasoning (Complete Geometry Course Lesson 2)

I’m just beginning Geometry. How can I start to talk about shapes?

Geometry is essentially the study of logical thinking and reasoning, using shapes as objects to test the reasoning. The word Geometry comes from two Greek roots: geo- “earth” and –metria “measuring”. This tells us that the earliest mathematicians used math to solve real-world problems, so they called this type of math, “measuring the world.” The geometer Euclid is given credit for establishing many of the ideas and practices that you will study in Geometry.

We can start by examining the smallest and simplest shapes in Geometry.

Points

A point is the smallest “shape” that we will use in Geometry. I put the word shape in quotes because a point is define as having no length and no width. You can think of a point as the smallest possible place that anybody can imagine. A point is so small that it cannot be made any smaller. Sometimes, we refer to points as having no dimension or being a zero-dimensional object.

In Geometry, we draw points as dots and name them with capital letters. For example, Point B could be drawn this way:

What if I only have part of a line? What do I call it, then?

The real world is not obviously made of straight lines and perfectly flat planes, since no object on earth can extend infinitely. Whenever we build anything or draw anything, we tend to use parts of a line. We know that these are only parts of a line because they end.

We can divide a line into two types of smaller parts: Rays and Segments.

Rays

Rays are parts of lines that end on one side, but go on forever on the other side. We draw them as lines that have arrows on one end. We name rays by the point at the end and another point closer to the arrow. For example, this ray

What kinds of common-sense statements can I make about lines, points, and planes? We can state some Postulates or Axioms about many fundamental concepts in Geometry. A postulate or an axiom is an accepted statement of a basic fact. These statements usually come from self-evident logic, or common sense. Postulate 1-1: There is exactly one line that passes through two specific points. – In Geometry, we use the word exactly to mean that there are no more and no fewer possibilities. Postulate 1-2: If two distinct lines intersect, then they intersect in exactly one point. – In Geometry, we use the word distinct to mean “different and unique.” – The word intersect means that the figures cross each other and have one or more points in common. Postulate 1-3: If two distinct planes intersect, then they intersect in exactly one line. Postulate 1-4: There is exactly one plane that goes through any three noncollinear points.

What kinds of common-sense statements can I make about lines, points, and planes?

We can state some Postulates or Axioms about many fundamental concepts in Geometry. A postulate or an axiom is an accepted statement of a basic fact. These statements usually come from self-evident logic, or common sense.

Postulate 1-1: There is exactly one line that passes through two specific points.

– In Geometry, we use the word exactly to mean that there are no more and no fewer possibilities.

Postulate 1-2: If two distinct lines intersect, then they intersect in exactly one point.

– In Geometry, we use the word distinct to mean “different and unique.”

– The word intersect means that the figures cross each other and have one or more points in common.

Postulate 1-3: If two distinct planes intersect, then they intersect in exactly one line.

Postulate 1-4: There is exactly one plane that goes through any three noncollinear points.

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