Slide 1 : 1/14/2010 1 1/14/2010 1 Welcome to
Class 2
geometry
Slide 2 : Logical Foundations of Geometry
Class 2
Rays & Angles Based on the Book
Logical Foundations of Geometry
by Late Prof. P. C. Joseph 1/14/2010 2
Slide 3 : 1/14/2010 3 Let’s Recollect A B segAB rayAB rayBA lineAB
Slide 4 : 1/14/2010 4 Visualizing a Plane
Slide 5 : 1/14/2010 5 l m A n p Lines & Planes
Slide 6 : Definition : The set of points between A and B, is called the interior of segAB. If segAB is open, its interior is segAB itself.
Definition : A set of points S is said to be convex , if for any two points A, B in S, segAB is a subset of S. Convex Set, Half-planes,
Angle 1/14/2010 6 A B A B
Slide 7 : 1/14/2010 7 Axiom : A line in a plane divides the plane, excluding the line, into two disjoint subsets such that1) each subset is a convex set, and2) if P is in one of the subsets and Q in the other, then segPQ intersects the line.
The two subsets are called half-planes Half-planes
Slide 8 : 1/14/2010 8 P Q l Line &
Half-planes
Slide 9 : Points in Half-planes Theorem: If three non-collinear points A, B, C and line l are in a plane such that the line does not contain any of the points , then either the line intersects two of the three segments AB, BC, CA or it does not intersect any of them. l
Slide 10 : 1/14/2010 10 Theorem The intersection of two convex sets is a convex set.
Slide 11 : 1/14/2010 11 Definition: Two rays with the same initial point are said to be co-initial. The union of two co-initial rays is called an angle. A B C AngleBAC
Slide 12 : Interior of an Angle
Slide 13 : 1/14/2010 13 Why is
the interior of an angle
convex ?
Is an angle convex ?
Slide 14 : 1/14/2010 14 Definition: A triangle is the union of the three segments joining
three non-collinear points. A B C Definition: The interior of a triangle is the intersection of
the interiors of its angles.
Slide 15 : 1/14/2010 15 Quadrilateral Definition: Suppose A, B, C, D are four points of which no three are collinear. Then the union of the segments AB, BC, CD, and DA is a quadrilateral provided no two of them intersect except at the end-points.
Slide 16 : 1/14/2010 16 (a) (b) (c) Which of the figures (a), (b), (c) are quadrilaterals?
Slide 17 : 1/14/2010 17 1: How many segments are there joining pairs of four points, no three of which are collinear? 2: How many segments are there joining pairs of five points, no three of which are collinear? 3: How many segments are there joining pairs of n points, no three of which are collinear?
Slide 18 : 1/14/2010 18 Please do not Quit
Before
writing
your comments