1. Give the converse and the contrapositive of the following statement: If the l
ID: 2899349 • Letter: 1
Question
1. Give the converse and the contrapositive of the following statement: If the lines l and m are parallel, tehn a transversal t to lines l and m cuts out congruent alternate interior angles. (Note: a transversal to a family of lines is a line which is not parallel to any of them, i.e., it shares an intersection point with each one of them.)
2. Give a formal proof to Proposition 2.5 using only the axioms for incidence geometry and the results we have proved already: For every point P there exist at least two distinct lines through P.
3. In each of the following interpretations of the undefined terms, which of the axioms of incidence geometry are satisfied and which are not? Tell whether each interpretation has the elliptic, Euclidean or hyperbolic parallel property. (Elliptic: there are no parallel lines. Euclidean: for a line l and a point P not on l, there is a unique line through P which is parallel to l. Hyperbolic: for a line `l and a point P not on l, there are at least two lines through P which are parallel to l. (Just brief justifications are sufficient as answers to these questions.)
(a) “Points” are lines in Euclidean three-dimensional space, “lines” are planes in Euclidean three-space, and “incidence” is the relation of a line lying in a plane.
(b) Same as in the previous part, except that we restrict ourselves to lines and planes that pass through a fixed point O.
(c) Fix a circle in the Euclidean plane. Interpret “point” to mean a Euclidean point inside the circle, interpret “line” to mean chord of the circle, and let “incidence” mean that the point lies on the chord. (Note that a chord of a circle is a segment whose endpoints lie on the circle.)
4. The following whimsical syllogisms are by Lewis Carroll. They are intended to illustrate that logical syntax depends only onthe form of the argument, not on the meaning or truth of the statements. Which of them are correct statements? (No argument needed.)
(a) No frogs are poetical; some ducks are unpoetical. Hence, some ducks are not frogs.
(b) Gold is heavy; nothing but gold will silence him. Hence, nothing light will silence him.
(c) All lions are fierce;some lions do not drink coffee. Hence, some creatures that drink coffee are not fierce.
(d) Some pillows are soft; no pokers are soft. Hence, some pokers are not pillows
Explanation / Answer
1. If the lines l and m are parallel, tehn a transversal t to lines l and m cuts out congruent alternate interior angles.
Converse: If a transversal t to the lines l an m cuts out the congruent alternate interior angles (equal alternate interior angles) then the lines l and m are parallel.
Contrapositive: If a transversal t to the lines l and m does not cut out congruent interior angles, then the lines l and m are not parallel.
4. a) none of the frogs are poetical, and only some ducks are unpoetical. Therefore the remaining ducks might be poetical and are not frogs. Therefore it is true.
b) True.
c)False
d) True
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