MATH 10 MODULE 6

• Euclidean Geometry


  The study of flat space, also called plane geometry (the study of lines and shapes) on a flat surface
• Euclidean Geometry


  • Cannot describe all physical space such as curved space (this is covered by non-euclidean geometry)
• The shortest distance between two points is a straight line
• The angles in any triangle add to 180 degrees
• A perpendicular line intersects another line at 90 degrees
• Five Axioms of Euclidean Geometry

  • Things that are equal to the same thing are equal
  • If equals are added to equals, then the whole are equal
  • If equals are subtracted from equal, then the remainders are equal
  • Things that coincide with one another are equal to one another
  • The whole is greater than the part
• Five Postulates of Euclidean Geometry

  • It is possible to draw a straight line from any point to any point
  • If you have a straight line, it is possible to extend in any direction to infinity
  • It is possible to draw a circle given any center and a radius
  • ALL right angles are equal or congruent
  • If you have two straight lines, and a third line crossing them, and the sum of the interior angle measure of the two initial lines is less than 90 degrees, then if you extend the lines, they will eventually cross on that side at one point
• Congruence


  Same size and shape
• Similarity


  Similar in shape but can be different size
• Congruence Criteria


  • SSS: Three corresponding sides are congruent
  • SAS: Two corresponding sides and the angle between them are congruent
  • ASA: Two corresponding angles and the side they include are congruent
• If two triangles have the same values for their corresponding angles, they are similar BUT not necessarily congruent
• Hyperbolic Geometry


  Characteristic postulate: Through a point P not on a line I, there exists at least two lines passing through point P parallel to line l
• Hyperbolic Geometry


  • Arises from negating Euclid's 5th Postulate (play fair's axiom)
  • Lines in this model are either diameters or arcs of circles intersecting the disk at right angles
  • In hyperbolic geometry, the sum of angles in a triangle is less than 180 degrees, and similar triangles are congruent
• Elliptic Geometry


  Characteristic Postulate: Through a point P not on a line l, there exist no line passing through point P parallel to line l
• Elliptic Geometry


  • Arises from negating Euclid's 5th postulate
  • Sum of interior angles of an elliptic triangle is more than 180
  • Similar triangles are congruent
  • Used in sea and air navigation for calculating distances and planning flight paths
• Projective Geometry


  The mathematical study of perspective and vanishing points, and helps in the creation of 3D images
• Desargues' Theorem


  If two triangles are perspective from a point, then they are perspective from a line
• Principle of Duality & Desargues' Theorem


  If two triangles are perspective from a line, then they are perspective from a point
• Topology


  • In topology, shapes are being dealt with as something that is made of a squishy rubber
  • If one geometric can be continuously transformed into another, then two objects are to be viewed as being topological the same
• Topological Space


  A non-empty set X with a topology T that satisfies:

  1) X is in T,

  2) The union of any member of T is also a member of T,

  3) The finite intersection of any member of T is also a member of T
• In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid: In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point
• The Möbius strip, also called the twisted cylinder, is a one-sided

  surface with no boundaries.
• Non-orientable - perplexing property of the Möbius
• Homeomorphism is the notion of equality in topology.
• The homeomorphism forms an equivalence relation on the class of all topological spaces

  Reflexivity: X is homeomorphic to X

  Symmetry: If x is homeomorphic to Y, then Y is homeomorphic to X

  Transitivity: If X is homeomorphic to Y, and Y is homeomorphic to Z, then X is homeomorphic to Z

  
  

  The resulting equivalence classes are called homeomorphism classes
• Two triangles are perspective from a line (the axis of perspectivity), if their sides can be put into a one-to-one correspondence in such a way that the axis of perspectivity is concurrent with each pair of corresponding sides
• When a transversal intersects two parallel lines, the interior angles formed on the same side of the transversal are called consecutive interior angles. These angles always add up to 180 degrees. So, if the interior angles are less than 180 degrees, it means they meet on the same side of the transversal.
• A transversal is a line that intersects two or more other lines. When it crosses a pair of parallel lines, it creates various angles where the lines intersect.
• When the lines are not parallel, the interior angles formed by the transversal on the same side may not necessarily add up to 180 degrees. In that case, the angles could be greater or less than 180 degrees depending on the specific configuration of the lines and the transversal