Problem Solving with Linear Models PDF

GOOD MORNING MATH - 2 Direction: The classroom will divide by two (2) groups. Under your chairs, there are puzzles you need to fix. Each group will analyze and give 3 only words that are related to the puzzle. LEARNING OUTCOMES Define and understand the concept of parallel lines and their pro...

GOOD MORNING MATH - 2 Direction: The classroom will divide by two (2) groups. Under your chairs, there are puzzles you need to fix. Each group will analyze and give 3 only words that are related to the puzzle. LEARNING OUTCOMES Define and understand the concept of parallel lines and their properties in Euclidean geometry (V). Use the Parallel Postulate and its equivalent forms to prove theorems and solve problems related to parallel lines and angles. Explain the significance of the Parallel Postulate in the development of Euclidean geometry EQUIVALENCE OF PARALLEL POSTULATES PLAYFAIR’S AXIOM -At most one line parallel to the given line can pass through the given point, if the point is not on the line. THE ANGLE SUM THEOREM FOR TRIANGLES -The sum of the angles in any triangle is THE EXISTENCE OF SIMILAR TRIANGLES -If two triangles have congruent corresponding angles, then the triangles are similar, meaning that their corresponding sides are proportional. EUCLID'S POSTULATE V -If two lines are intersected by a transversal in such a way that the sum of the degree measures of the two interior angles on one side of the transversal is less than 180, then the two lines meet on that side of the transversal. THEOREM 4.5 - EUCLID'S FIFTH POSTULATE IF AND ONLY IF HILBERT'S PARALLEL POSTULATE THE ISSUE OF EUCLID’S POSTULATES 5 Numerous mathematicians have tried to demonstrate the Fifth Postulate based on the other four postulates, aiming to prove it as a theorem rather than an axiom. These efforts frequently included examining alternative expressions of the postulate or investigating diverse geometric constructions. Nonetheless, all of these endeavors have ultimately been unsuccessful. The inability to demonstrate the Fifth Postulate resulted in the emergence of non- Euclidean geometries like hyperbolic and elliptic geometry. These geometries stem from different interpretations of the Parallel Postulate, resulting in distinct and captivating characteristics that diverge from those of Euclidean geometry. The Fifth Postulate of Euclid remains an intriguing and contentious subject in mathematics. Despite sparking extensive discussions and research, it remains acknowledged as an axiom. The emergence of non-Euclidean geometries has, however, demonstrated that there are alternative methods to approach geometry, broadening our comprehension of the fundamentals of mathematics. PROPOSITION 4.7. HILBERT'S PARALLEL POSTULATE IF AND ONLY IF, IF A LINE INTERSECTS ONE OF TWO PARALLEL LINES, THEN IT ALSO INTERSECTS THE OTHER. PROPOSITION 4.8. HILBERT’S PARALLEL POSTULATES IF AND ONLY IF CONVERSE TO THEOREM 4.1 (ALTERNATE INTERIOR ANGLES). PROPOSITION 4.9 HILBERT’S PARALLEL POSTULATE IF AND ONLY IF, IF T IS A TRANSVERSAL TO L AND M, L IS PARALLEL TO M, AND T PERPENDICULAR TO L, THEN T PERPENDICULAR TO M. PROPOSITION 4.10. HILBERT’S PARALLEL POSTULATE IF AND ONLY IF, IF K IS PARALLEL TO L, M PERPENDICULAR TO K, AND N IS PERPENDICULAR TO L, THEN EITHER M = N OR M IS PERPENDICULAR TO N.  PROPOSITION 4.11. IF HILBERT’S PARALLEL POSTULATE THEN, THE ANGLE SUM OF EVERY TRIANGLE IS 180 DEGREE. REFERENCE Mathspace. "11.04 Problem solving with linear models | Grade 8 Math | Common Core 4 - 2022 Edition." Accessed 22 July 2023, https://mathspace.co/textbooks/syllabuses/Syllabus- 1157/topics/Topic-21913/subtopics/Subtopic-279958/? searchString=&activeTab=theory