1. Euclidean vs Non-Euclidean Geometry
What’s the difference?

2. 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 //.

3. 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.

4. 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

5. Elliptic vs Euclidean geometry:

6. Hyperbolic geometry
No similar triangles exist.

7. 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

8. 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!

9. Which of the following mathematicians is considered to be the “Father of Geometry”?
a. Archimedes c. Pythagoras
b. Plato d. Euclid

10. One of the reasons we study geometry is to develop logical thinking
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 c. Greeks
b. Egyptians d. Mayans

11. Which mathematician gave us a very important rule about right triangles?
a. Euclid c. Archimedes
b. Pythagoras d. Descartes

12. Which of Euclid’s postulates separate his theory of geometry from all non-Euclidean geometry?
a. parallel postulate c. protractor postulate
b. ruler postulate d. distance formula

13. Which of the following statements is true only in non-
Euclidean 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.