Michael Bonomi Understand the principles and properties of axiomatic (synthetic) geometries (0016) Euclidean Geometry: To understand this part of the CST I decided to start off with the geometry we know the most and that is Euclidean:
− Euclidean geometry is a geometry that is based on axioms and postulates − Axioms are accepted assumptions without proofs − In Euclidean geometry there are 5 axioms which the rest of geometry is based on − Everybody had no problems with them except for the 5 axiom the parallel postulate − This axiom was that there is only one unique line through a point that is parallel to another line − Most of the geometry can be proven without the parallel postulate − If you do not assume this postulate, then you can only prove that the angle measurements of right triangle are ≤ 180° Hyperbolic Geometry:
− We will look at the Poincare model − This model consists of points on the interior of a circle with a radius of one − The lines consist of arcs and intersect our circle at 90° − Angles are defined by angles between the tangent lines drawn between the curves at the point of intersection − If two lines do not intersect within the circle, then they are parallel − Two points on a line in hyperbolic geometry is a line segment − The angle measure of a triangle in hyperbolic geometry < 180° Projective Geometry:
− This is the geometry that deals with projecting images from one plane to another this can be like projecting a shadow − This picture shows the basics of Projective geometry − The geometry does not preserve length or angle measurement − The theorem of Menelaus of Alexandria is seen as the similarity theorem which is 퐶′퐷′/퐷′퐴′ = (퐶′퐸′/퐸′퐵′) ∗ (퐵′/퐴′) − This shows how the smaller image can equal the bigger image based on where the geometry is getting projected − This system is more abstract and based on formulas to help see the differences in sizes of the pictures
Comparing and Contrasting These Axiomatic Systems: Similarities:
− All these systems run on what are called undefined terms − These are terms that are so abstract yet simple that trying to define them exactly will lead to more confusion − These terms are terms such as point and line − A point in these geometries is a location with no size, such as width, depth, and length − A line is defined in all geometries and has no thickness − Both terms are common to be seen in all geometries and just taken to be true even though they are shown differently Differences:
− There are different types of parallel lines in each of the geometries − Angle measurements in hyperbolic are smaller than that of its Euclidean counterpart − In Hyperbolic unlike Euclidean two lines can have the same measure even though to the eye they look different For high school mathematics you can prove geometry using either t-chart proofs or paragraph proofs both equally acceptable. Properties of Lines and Angles to Characterize Geometric Relationships: In this section we will look at the properties of lines to see the relationship between two other lines. We will look at what it means for each term in Euclidian geometry to exist. This section will then lead into the concepts of similarity and congruence of geometric figures. Parallelism: Parallel Lines and special features of them
A B - The definition of parallel lines are lines on a plane that never meet. They are always the same distance apart - A Transversal is a line that passes through two parallel lines - A transversal that passes through two vertical lines will create alternate interior angles that are congruent - If two lines are cut by a transversal and the alternate interior angles are congruent then they are parallel A - Using symbols to show that line A is parallel to line B is written as A II B Perpendicularity: What perpendicular lines are B - Two lines are perpendicular to each other if they meet at a 90º or right angle - Using symbols to show that line A is perpendicular to line B is written as A⊥B Supplementary Angles: What it means if two angles are supplements of each other - Two angles are Supplementary when their measures add up to 180º - m<1 + m<2=180º - This is would be the measure of both sides of a line that cuts another straight lone - Complementary Angles: Are the same as supplementary angles but their measures add up to 90º - This is would be anywhere where a right angle is split into two pieces Vertical Angles: Where to find vertical angles - Vertical Angles are a pair of non-adjacent angles formed when two lines intersect - Vertical Angles are congruent to one another 1 3 and 2 4 - The addition of all vertical angles about one point will add up to 360º Bisectors: Both angle and line segment bisectors
- Angle Bisector: A line that splits and angle into two congruent parts - The angle bisector is usually a Ray that starts from the vertex of the angle it is bisecting and goes on infinitely - Bisects the side of the angle that is <180 - Perpendicular Bisector of a Line Segment: A line that is perpendicular to the line segment and splits it into two congruent parts - Usually a ray that starts at the Midpoint or middle of the line segment. Using the Properties to Help Explain Similarity and Congruence of Geometric Figures: This section is concepts about different types of geometric figure and how they are similar or congruent. Also, gives definitions of geometric figures. Triangle: Is a polygon with three edges and three vertices - Similarities of Triangles: Two triangles are similar if they have the same angle measurements but not necessarily the same side lengths. (Two triangles are congruent of they have the same shape) - If two triangles are similar the proportion of the corresponding sides are congruent - This is the same as saying each triangle is a scaled version of the other triangle - If a triangle is cut by a line that is parallel to one of the other sides, it creates similar triangles with a smaller triangle inside a larger triangle sharing one of its vertices - When a line cuts a triangle parallel to one side of the triangle it creates a proportion 푎푑 푎푒 of sides in which = 푐푑 푒푏 - Congruence of Triangles: Two triangles are congruent if both their sides and angles that correspond to each other are congruent - All congruent triangles are similar - In order to prove two triangles are congruent you need to know one of the following information’s: o Side, Side, Side (SSS): If all three sides of two triangles are congruent then the triangles are congruent to each other o Side, Angle, Side (SAS): If two sides and an angle in between both sides are congruent then both triangles are congruent o Angle, Side, Angle (ASA): Two triangles that have two angles with a side in between them congruent then the triangles are congruent o Angle, Angle, Side (AAS): Two triangles with two angles and a then the side after the angles congruent then the triangles are congruent o Hypotenuse, Leg (HL): Two Right triangles with a congruent Hypotenuse and Leg are both congruent o Side, Side, Angle (SSA): This is a special case where if the second side is longer or congruent to the first side and the dimensions given can create a triangle the two triangles are congruent Quadrilateral: A polygon with four edges and four vertices. We will look at 4 types. - Parallelogram: a quadrilateral with opposite sides parallel to each other - Opposite sides are congruent - Opposite angles are congruent - Consecutive angles are supplementary - If one angle is right, then all the angles are right - The diagonals bisect each other - Each diagonal separates the parallelogram into two congruent triangles o Rectangles: All rectangles are parallelograms with all the angles being right angles o All rectangles follow the same properties of parallelograms ▪ Squares: All squares are rectangles with all sides congruent to each other ▪ Follows all the properties of a rectangle - Trapezoids: is a quadrilateral with one pair of opposite sides parallel - Isosceles Trapezoids: Are trapezoids in which the non-parallel sides are congruent o The angles on either side of the base are congruent o The diagonals are congruent Circles: the locus of all points equidistant from a central point - Circles with different radii are similar - Circles with the same radii are congruent - The central angle which intercepts an arc is double any inscribed angle that intercepts the same arc - If the radius is perpendicular to a chord bisects the chord - Lines tangent to the circle are perpendicular to the radius - Two tangent lines drawn to the same point the line segments from the point of tangency to the end point are congruent Procedures used in Geometric Constructions: In this section we will look at some simple procedures on how to construct certain geometric properties. - To construct anything in geometry you need to base everything on the radii of circles - How to copy an angle using compass and straight edge: o Step 1: Draw line with point B o Step 2: Move the compass on ray A to any point name it point S o Step 3: Move the compass on ray B the same distance as line segment AS o Step 4: With the compass measure from point S to point T o Step 5: Without changing the compass do measure from point V to a point on the circle BV where the two circles cross draw a line to the point W Applying Geometric Principles to Analyze Three-Dimensional Figures: This section we will quickly discuss what properties from two-dimensional figures apply to three-dimensional figures. If we look at something and we do not know what type of object it is but are given some dimensions apply principles of two-dimensional figures can help us figure out the object. So, if the object has parallel bases with lines connecting each base at 90º angles then we know this is most likely some sort of prism. Now if we know that the base shape has 90º angles and opposite sides are congruent we know they base is a rectangular prism (can even be a cube which is just a special case for a rectangular prism). This is how we can apply our knowledge to every question to figure out the three-dimensional shape and other properties from it. Examples: Problem 1: Using these 3 Axioms known as incidence geometry answer the question. A(1) For any two Distinct points P and Q in S, there is a unique line containing them. A(2) Every line contains at least two points. A(3) There exist at least three non-collinear points. That is, there are three points that don’t lie on the same line. Let us consider the set S={a,b,c} and call the elements of S “points.” Call a subset of the two distinct elements a “line.” Show that S is an incidence geometry. In this geometry parallel lines are lines that have no points in common. What can we say about parallel lines in this model? Problem 2: Prove that two lines perpendicular to same line are parallel to each other. Problem 3: With a straight edge and compass construct a from point P a perpendicular bisector that is equidistant from two points on line L.
Problem 4: Given Parallelogram A,B,C,D and, diagonals 퐷퐵̅̅̅̅ and 퐴퐶̅̅̅̅ which meet at point E, prove that opposite sides are congruent and ̅퐴̅̅퐸̅ ≅ 퐶퐸̅̅̅̅ and ̅퐷퐸̅̅̅ ≅ 퐵퐸̅̅̅̅
E
Problem 5: Prove Two tangent lines drawn to the same point the line segments from the point of tangency to the end point are congruent. Problem 6: Explain why the only one case in which Side, Side, Angle does not prove that two triangles are congruent. Problem 7: Prove If three angles of one triangle equal to three angles of another triangle, then the corresponding side of the triangle are in proportion. Hence, they are similar triangles. Problem 8: Given a right triangle construct a cone. Explain why the base of the cone changes proportionally to the height as you go up the cone.
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