GEOMETRY POSTULATES AND THEOREMS

Corresponding angle hypothesis, or CA, when two parallel lines are cut in the horizontal direction, the corresponding angle is

If the inner angle theorem or two parallel lines are cut in the horizontal direction, substituting the AIA theorem, the alternate internal angle becomes as follows.

Alternate external angle theorem, or AEA theorem when two parallel lines are alternately cut in the horizontal direction

Theorem and hypothesis: Theorems and hypotheses are geometrically correct explanations. For example, all right angles are congruent, or all radii of a circle are congruent. The difference between the hypothesis and theorem is that the hypothesis is correct, but we need to prove that the theorem is correct based on the hypothesis and proved theorem. Unless you are writing a doctorate, this distinction is not something you need to care about. Papers on geometric deductive structure However, there is a possibility that you have not obtained a doctorate at the moment. In geometry, you should not sweat this detail

Book 1 covers important topics of planar geometry, including five hypotheses (including well-known parallel hypothesis) and Pythagorean theorem, equality of angle and area, parallelism, angle of triangle There are five common concepts. Combine various shapes and make up. Book 10 prove the irrationality of the square root of (for example) non-square integers and divide the square root of non-commutable lines into 13 disjoint categories. Euclid introduced the term "irrational" here, but it has a different meaning from the concept of contemporary irrational numbers. He also gave way to the production of Pythagoras triad.

Because the famous Pythagorean theorem depends on parallel hypothesis, it is the Euclidean geometry theorem. However, in chapter 5 and chapter 6 we meet non - Euclidean variants of this theorem and propose the Pythagorean theorem unified in Chapter 7. This is 7.4.7 which appeared recently. As a result, the Pythagorean theorem appears as Proposition 47 at the end of the first volume of the Euclidean element, and presents the Euclidean proof below. Pythagorean theorem, including Euclidean geometry and non-Euclidean geometry, is the foundation of the measurement system used in this paper. We also noticed that the proposition number 48, the last proposition of the first volume, gives the opposite of what the constructor uses. Measuring triangle feet \ (c ^ 2 = a ^ 2 + b ^ 2 \) The opposite \ (c \) is correct. Interested readers can find an online version of the Euclid element here.