GEOMETRYGEOMETRYProperties and relationships of points, lines, planes,

solids. Undefined elements: point, line, planeAxiom: self-evident truthPostulate: statement accepted as true without proofTheorem: statement which is proven2 lines are parallel iff they line on same plane but do not

intersect2 planes or a line & a plane are parallel if they do not

intersect2 lines are perpendicular/orthogonal if the adjacent

angles of their point of intersection are right anglesSkew lines: 2 lines not coplanar/non-parallel/non-

intersection

EUCLIDEAN GEOMETRYEUCLIDEAN GEOMETRYElementsDeductive ReasoningPostulates:1. Two points determine a line2. A line can be extended indefinitely in both directions3. A circle may be drawn with any given center and

radius4. All right angles are equal5. Parallel Postulate Given a point P not on a line L, then there is one and

only one line (L2) passing thru P and parallel to L

EUCLIDEAN GEOMETRYEUCLIDEAN GEOMETRYElementsDeductive ReasoningPostulates:1. Two points determine a line2. A line can be extended indefinitely in both directions3. A circle may be drawn with any given center and

radius4. All right angles are equal5. Parallel Postulate Given a point P not on a line L, then there is one and

only one line (L2) passing thru P and parallel to L

HYPERBOLIC GEOMETRYHYPERBOLIC GEOMETRY

Karl Friedrich Gauss Nicholas Lobachevski Johann BolyaiDisc Model : Hyperbolic planeFundamental Circle: Fixed circle C, center at OHyperbolic points: interior to CHyperbolic lines: circular arc perpendicular to C

HYPERBOLIC GEOMETRYHYPERBOLIC GEOMETRY

Disc Model : Hyperbolic planeFundamental Circle: Fixed circle C, center at O

(Poincare Disc)Hyperbolic points: interior to circle CHyperbolic lines: circular arc perpendicular to circle

C

ELLIPTIC ELLIPTIC GEOMETRYGEOMETRYBernard Riemman

Spherical/Earth ModelElliptic Points: All points ON SURFACE of sphereElliptic Lines: Great circles (diameter = equator)Antipodal point

ELLIPTIC ELLIPTIC GEOMETRYGEOMETRYBernard Riemman

Spherical/Earth ModelElliptic Points: All points ON SURFACE of sphereElliptic Lines: Great circles (diameter = equator)Antipodal point

COMPARISONCOMPARISONEuclidean Hyperbolic Elliptic

Any 2 lines intersect in

1 point 1 point 1 point/s

Parallel lines are equidistant Converge in 1 direction; diverge in the other

DNE

If a line intersect 1 of 2 parallel lines, it must

Intersect the other

May or May not intersect the other

NA

2 lines perpendicular to a line are

parallel parallel Intersecting

Sum of angles in a triangle

180 < 180 > 180

Area of triangle Independent of sum of the angles

Proportional to the deficit of sum

Proportional to the excess of sum

2 triangles with corresponding angles congruent

similar congruent congruent

COMPARISONCOMPARISONEuclidean Hyperbolic Elliptic

Sum of angles in a triangle

180 < 180 > 180

PROJECTIVE PROJECTIVE

GEOMETRYGEOMETRY Johannes KeplerVictor PonceletPrinciple of Perspectivity: applied on a 2D canvas

to depict 3DA 1-1 correspondence Ideal points, ideal line, ideal plane

PROJECTIVE GEOMETRYPROJECTIVE GEOMETRYPrinciple of Perspectivity

PROJECTIVE GEOMETRYPROJECTIVE GEOMETRYPrinciple of Perspectivity: applied on a 2D canvas

to depict 3D

TOPOLOGYTOPOLOGYFigures (mathematical spaces) whose geometric

properties are unchanged by continuous deformation (stretching, twisting, shrinking)

Topological transformation : continuous deformation

Properties preserved: outside/insideNot preserved: shape, magnitudeTopologically equivalent: objects changed into

another by a topological deformation

TOPOLOGYTOPOLOGYTopologically equivalent: objects changed into

another by a topological deformation

TOPOLOGYTOPOLOGYTopologically equivalent: objects changed into

another by a topological deformation

TOPOLOGYTOPOLOGYTopologically equivalent: objects changed into

another by a topological deformation

EUCLIDEAN GEOMETRYEUCLIDEAN GEOMETRYI. Triangle Properties a) sum of any 2 sides of a triangle is greater than the 3rd

side 2,3,6 2,3,5 2,3,4 b) sum of measures of the angle of a triangle is 180

c) congruence: same shape, same measurement SAS ASA SSS d) similarity: same shape e) area and perimeter rectangle: A = l x w P = 2l + 2w triangle: A = (1/2)bh P = add all 3 sides circle: A = 𝜋r2 C = 2𝜋r

TRIGONOMETRYTRIGONOMETRYRight triangleSOHCAHTOAsine A = sin A = side opposite/hypothenusecosine A = cos A = side adjacent/hypotenuse tangent A = tan A = sin A/ cos A = side opposite/

side adjacentcosecant A = csc A = 1/sin Asecant A = sec A = 1/cos Acotangent A = cot A = 1/tan A = cos A /sin AAngle of elevation/depression

TRIGONOMETRYTRIGONOMETRYAn airplane 405 ft above a landing field when the

pilot cuts out his motor. He glides to a landing at an angle of 130 with the field. How far will he glide in reaching the field ?

A road running from the bottom of the hill to the top is 625 m. If the hill is 54 m high, what is the angle of elevation of the road ?