Metromatematicas Axioms Proofs Logical Approach.pptx

For the good of humanity and the blue planet, first the evolution of the 3 Natural Intelligences(3iN – Rational, Emotional and Instinctive) of the predictor brain of children, adolescents and young students.

1. MetroMatematicas

2. Introduce the deductive method Understand types of Geometry Be able to reason quantitatively Use properties of rational numbers Know precise definitions of point, line, ray, line segment, and distance along a line Prove theorems about lines 2 Objectives

3. Definitions Axiom – proposition so simple and evident that is accepted without demonstration. Ex: the whole is greater than the part. Postulate – a proposition as evident as an axiom. Therefore, it is also accepted without demonstration. Ex: there are infinite points. Theorem – a proposition that can be demonstrated. The demonstration includes a group of arguments that leads to evidence of truth of the proposition. In wording of any theorem two parts are distinguished: The hypothesis, which is the supposition, and the thesis, which is what has to be demonstrated. Ex: the sum of the interior angles of a triangle is equal to two right angles.

4. Definitions Corollary – a proposition that is deducted from a theorem as consequences of the theorem.. Ex: From the theorem: the sum of the interior angles of a triangle is equal to two right angles, the following corollary is deducted: the sum of two acute angles of a right triangle is equal to a right angle. Reciprocal Theorem – all theorems have a reciprocal, the hypothesis of the reciprocal is, respectively, the thesis and the hypothesis of the other theorem, which, in this case, is called direct theorem. Ex: The reciprocal theorem of “ the sum of the interior angles of a triangle is equal to two right angles”, says that: If the sum of the interior angles of a polygon are equal to two right angles, then the polygon is a triangle.

5. Definitions Lemma – a proposition that serves as the base for the demonstration of a theorem. It’s similar to a preliminary theorem that leads to another theorem, which is considered more important. Ex: To demonstrate the volume of a pyramid, the lemma that says: A triangular prism can be decomposed into three equivalent tetrahedrons. Currently, the word lemma is not used extensively and it is often called theorem, or preliminary theorem. Point – a point is that which has no part. Geometric points are so small that have no dimensions. I submit the following postulate: there are infinite points. Points are often designated with upper case letters, and are represented with a dash, a small circle or a cross. See Fig 1 below:

6. Definitions Straight line – a line is breadtheless length. Illustrations of this group include: a luminous beam and the edge of a ruler. There is exactly one line passing through two distinct points. Two distinct lines cannot have more than one point in common. Often, the straight line is designated by two points with the line symbol  over them. For example, the straight line AB in Fig 2 is represented as AB.

7. Definitions Ray – if we mark a point A on the ray, the group of points formed by the A, from all of those that precede and follow it, is called a ray. Point is the origin of the ray. Often a ray is represented by an origin and another point in it with the symbol  over. See Fig 5 Line Segment – on a straight line we mark two points, A and B, then a segment is the group of point that is between A and B, as well two additional point named ends. Usually, the point named first is called the origin, whereas the other point is called the end. The following postulate is admitted: The shortest distance between any two points is the line segment joining them. A segment is designated by letters at it’s ends with a dash line above them.

8. Theorem 1 Two convex broken lines ABCD and AFED share the same endpoints and lie on the same side of the line connecting the endpoints, so AFED is contained inside ABCD, then the sum of the segments in ABCD is greater than the sum of the segments AFED.

9. Theorem 1 Hyphotesis ABCD upper convex broken line. AFED lower convex broken line. A and D common endpoints. Thesis AB + BC + CD > AF + FE + ED Auxiliary construction. Let’s stretch out AF up to intersect BC in G and to FE up to intersect CD in H.

10. Theorem 1 Demonstration In ABGF: AB + BG > AF + FG (1) In FGCHE: FG + GC + CH > FE + EH (2) In EHD: EH + HD > ED (3) Postulate of the shortest distance between two points. Adding in an orderly manner (1), (2), and (3), we have: AB + BG + FG + GC + CH + EH + HD > AF + FG + FE + EH + ED (4) But: BG + GC = BC (5) CH + HD = CD (6)

11. Theorem 1 Replacing (5) and (6), in (4), we have: AB + BC + CD + FG + EH > AF + FG + FE + EH + ED (7) Simplifying: AB + BC + CD > AF + FE + ED