Download M 333 L Fall 2002 Solutions to Homework #2

Homework #3A, Parts II and III Assignment
Homework #3A, Part II Assignment:
M 333 L
Spring 2017
Exercises 3.2 : # 9
Hints and Observations Regarding #9 of Exercises 3.2
1. In Theorem 3.2.8, the statement proven in the text is:

"IF a point is P equidistant from the endpoints of a line segment,
THEN it is on the perpendicular bisector of that line segment."
2. The Converse of the statement proven in the text is:

"IF a point P is on the perpendicular bisector of a line segment
that point P is equidistant from the endpoints of that line segment
, THEN
."
3. The assignment in exercise #9 is to prove the converse statement.
(You can use this converse statement above in the "To Prove:" heading of your solution.)
4. The proof of Theorem 3.2.8 presented in class (and in the handout "Introduction to
Neutral Geometry") is divided into two cases:

Case 1 assumed that the point P was on the line AB and

Case 2 assumed that the point P was not on the line
AB .
5. In proving the converse, you must also divide the proof into two cases:
Case 1, which assumes that P is on line segment AB
(and thus is the midpoint of line segment AB ) and
Case 2, which assumes that P is not on the line segment AB
(and so some other point, say M, is the midpoint of line segment AB ,
but P still on the perpendicular bisector).
Remember that the proof of an IF-THEN statement must include the assumption that the IF part is
true, either by explicitly saying "Suppose" or "Assume" or by describing a fully general context in
which the IF part is true..
Homework #3A, Part III Assignment
It is given that lines l and m are intersected by a transversal t at points P and Q . Points A,
B, C, D, E, and F are defined as indicated in the figure below.
It is also given that APQ  PQF .
To Prove:
Lines l and m do not intersect at any point in the half-plane side of line t containing C and F.
Proof:
Suppose, by way of contradiction, that lines l and m intersect at some point, say at point T, in
the half-plane side of line t containing C and F.
t
l
Q
D
E
F
T
m
P
A
C
B
[ Hint: Apply the Exterior Angle Theorem (Theorem 3.2 9) to ∠ APQ. ]
[Note: You are not allowed to apply the Alternate Interior Angles Theorem (in case you have
wisely been reading ahead. First of all, we haven't proved it yet so it is not available.
But also, this exercise is the meat of the proof of the Alternate Interior Angles Theorem
and we cannot apply a particular theorem in the proof of that same theorem.]