Transcript Euclidean vs Non-Euclidean Geometry

Euclidean vs NonEuclidean
Geometry
What’s the difference?
Consider two straight lines indefinitely extended in a
two-dimensional plane that are both perpendicular to a third line:
In Euclidean geometry, these two lines must be //.
In Euclidean geometry the lines remain at a constant distance
from each other even if extended to infinity, and are known as
parallels.
In hyperbolic geometry they "curve away" from each other,
increasing in distance as one moves further from the points of
intersection with the common perpendicular; these lines are often
called ultraparallels.
In elliptic geometry the lines "curve toward" each other and
 eventually intersect.
Non-euclidean geometry can be understood by picturing the
drawing of geometric figures on curved surfaces, for example,
the surface of a sphere or the inside surface of a bowl.
 // lines
1 & only 1 // line
No // lines
Elliptic vs Euclidean geometry:
Hyperbolic geometry
No similar triangles exist.
Another major difference between the three geometries is
the sum of the angles of a triangle.
In Euclidean, the sum = 180º
In Hyperbolic, the sum < 180º
In Elliptical, the sum > 180º
In high school geometry, our main focus will be on
Euclidean geometry. You will learn more about the
other geometries in Pre-Cal and Calculus.
BTW, there are many more aspects of geometry than
the three listed here. See YouTube for some very
interesting videos!
Which of the following mathematicians is considered
to be the “Father of Geometry”?
a. Archimedes
b. Plato
c. Pythagoras
d. Euclid
One of the reasons we study geometry is to develop
logical thinking. Which of the following groups of
people is most responsible for including logical
reasoning in its investigations of geometric
concepts?
a. Babylonians
b. Egyptians
c. Greeks
d. Mayans
Which mathematician gave us a very important rule
about right triangles?
a. Euclid
b. Pythagoras
c. Archimedes
d. Descartes
Which of Euclid’s postulates separate his theory of
geometry from all non-Euclidean geometry?
a. parallel postulate
b. ruler postulate
c. protractor postulate
d. distance formula
Which of the following statements is true only in nonEuclidean geometry?
a. The sum of the measures of the three angles of a
triangle could be more or less than 180°.
b. The sum of four angles of a quadrilateral is equal
to 360°.
c. Parallel lines are always equidistant.
d. Parallel lines have no points in common.