Euclidean vs Non-Euclidean Geometry

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Presentation transcript:

Euclidean vs Non-Euclidean 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 c. Pythagoras b. Plato d. Euclid

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

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

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

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.