After completing this tutorial, you will be able to complete the following:
What are propositions, axioms, and postulates?
~ A proposition is a declarative sentence that is either true or false. An axiom is a proposition that is assumed to be true; we do not prove axioms. In geometry, axioms are sometimes called postulates; we do not prove postulates.
What are the statements in Euclid's five postulates?
~ Euclid's postulates are as follows:
Postulate 1: One and only one line passes through two different points.
Postulate 2: A line segment can be extended infinitely.
Postulate 3: A circle whose center and radius are known can be drawn.
Postulate 4: All right angles are congruent.
Postulate 5: Given a line and a point not on the line, there is only one line through the
point that is parallel to the first line.
Juan wants to prove that an equilateral triangle can be constructed with sides whose lengths are the length of a given line segment. The proof requires Juan to construct circles with known centers and radii. Does Juan need to prove that the required circles exist?
~ No, Juan does not need to prove that the required circles exist. Since the centers and radii of the circles are known, Euclid's third postulate guarantees that the circles can be drawn.
Are all five of Euclid's postulates equally important for Euclidean geometry, or is one postulate more important than the others?
~ Answers will vary. Strictly speaking, all five axioms are required for Euclidean geometry. Changing any axiom results in a different geometrical system. A case can be made that changing the fifth postulate provides some of the more interesting geometrical systems.
In your response to question 2 in the previous section, did you list any of Euclid's postulates?
~ Answers will vary. This may provide an opportunity to illustrate the difference between postulates, definitions, and theorems or propositions
|Approximate Time||2 Minutes|
|Pre-requisite Concepts||Students should be familiar with the following terms:|
|Type of Tutorial||Animation|
|Key Vocabulary||Euclidean postulates, postulates, points|