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Geometry: High School15 chapters | 160 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Learn how the way we do proofs in geometry had its start with Euclid in this video lesson. Learn about his contributions to the geometry we know today. Also learn about the five basic truths that he used as a basis for all other teachings.

Euclid was a Greek mathematician who introduced a logical system of proving new theorems that could be trusted. He was the first to prove how five basic truths can be used as the basis for other teachings. He wrote a series of books that, when combined, becomes the textbook called the *Elements* in which he introduced the geometry you are studying right now. It is in this textbook that he introduced the five basic truths or postulates upon which the whole of geometry at that time was based.

What are these postulates that he introduced as the basis for geometry? You might be surprised by how simple some of them seem to be. And that is as it should be. These are basic truths that should be self-evident and do not require formal proofs. The five postulates that he introduced are these:

1. A line can be drawn between any two points.

2. Any line segment can be extended to infinity in both directions.

3. A circle can be described with just a center point and radius.

4. A right angle is equal to all other right angles.

5. This last one is also called the parallel postulate. It states that if one line intersects two other lines and forms angles less than 90 degrees on one side, then the two lines will intersect on that side.

Looking at the first four, you can see that these are intuitive and very simple truths. You can see that they are true and do not require a formal proof. The last one may not seem so intuitive, and mathematicians have argued that the parallel postulate requires a proof, but mathematicians have seen over time that it is a basic truth that does not need a formal proof.

These five postulates, or basic truths, are also called axioms, and Euclid used these axioms to introduce his axiomatic system.

The axiomatic system is a system of formal proofs. It provides a system where new theorems are proved using basic truths. Because these theorems are based on basic truths, they can be relied on and used with trust that they will work and will give reliable answers.

How does the axiomatic system work? You can think of it as a lawyer trying to prove a case. A lawyer needs reliable evidence that cannot be refuted. A lawyer needs to review some basic facts that he knows are true, and then he builds on these facts to build new ones that lead to his conclusion of either guilty or not guilty. It is similar with the axiomatic system. A set of basic truths that are self-evident are used to prove new conclusions. Mathematicians use this system to either show a new conclusion or new theorem to be true or false. True theorems are then presented to their students or written in textbooks. All of these theorems that have been proved are now part of what we know as geometry. Many of the theorems you will learn have been proved using the axiomatic system.

The part of geometry that uses Euclid's axiomatic system is called Euclidean geometry. For thousands of years, Euclid's geometry was the only geometry known. But in the nineteenth century, other geometric spaces and ways of thinking were introduced. So, now we have Euclidean geometry and non-Euclidean geometry. What is the difference? As you can see from the basic truths, Euclidean geometry assumes that lines and surfaces are straight and flat. Non-Euclidean geometry, on the other hand, includes lines and surfaces that bend. Why do we need to know geometry where lines and surfaces can bend? Because our Earth is round and if we did math on its surface, we would need to account for that bend. So now, you will come across the teachings based on Euclid's geometry that assumes flat surfaces as well as other geometries that use curved spaces. Both are taught today.

What have we learned in this video lesson? We've learned that Euclid made a huge contribution to the study of geometry by writing his series of books that became the textbook called *Elements*. In it, he introduced five basic truths or axioms.

These are that:

- Lines can be drawn connecting any two points.
- Line segments can be extended infinitely in both directions.
- Circles can be described by a center point and radius.
- All right angles are equal.
- Two lines will intersect on the side where a third line intersects the two lines and makes angles less than 90 degrees.

These basic truths become the basis for Euclid's axiomatic system. The axiomatic system is a system where new theorems are proved by the five basic truths. Because these new theorems can be proved using basic truths, these new theorems can be thought of as true and used in calculations and to solve more problems. Today, we study Euclidean geometry that defines lines and surfaces as straight and flat along with non-Euclidean geometry that deals with curved lines and surfaces.

After you've completed this lesson, you'll be able to:

- Explain the impact that Euclid's
*Elements*had on geometry - List Euclid's five truths
- Describe Euclid's axiomatic system and how it applies to straight, flat lines as well as to curved lines and surfaces

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Geometry: High School15 chapters | 160 lessons

- What is Geometry? 4:36
- Inductive & Deductive Reasoning in Geometry: Definition & Uses 4:59
- Thales & Pythagoras: Early Contributions to Geometry 5:14
- The Axiomatic System: Definition & Properties 5:17
- Euclid's Axiomatic Geometry: Developments & Postulates 5:58
- Properties and Postulates of Geometric Figures 4:53
- Algebraic Laws and Geometric Postulates 5:37
- Go to High School Geometry: Foundations of Geometry

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