MTH 250 Graded Assignment 8, part one
Parallel lines in absolute and Euclidean geometries
Material from Kay, section 4.1
Q1:
Theorem 1 (Parallelism in Absolute Geometry) on p. 212 states that
If two lines in the same plane are cut by a transversal so that a pair of alternate interior
angles are congruent, then the lines are parallel.
There are several equivalent versions of this theorem (or corollaries, if you prefer), including
(1) If two lines in the same plane are cut by a transversal so that a pair of corresponding
angles are congruent, then the lines are parallel.
(2) If two lines in the same plane are cut by a transversal so that a pair of same side interior
angles are supplementary, then the lines are parallel.
Prove the corollaries.
Proof of (1):
Given: Suppose that l and m lie in the plane and are cut by transversal t, forming the numbered
angles as shown. Assume betweenness as is apparent. Suppose also that a pair of corresponding
angles are congruent – say 1  5 (since the chosen pair of angles is arbitrary, a similar proof
would hold for any pair of corresponding angles we choose, so there's no sense in considering all the
cases).
Statement/Reason: Since 1 and
supplementary. Therefore
4 form a linear pair, by the Linear Pair Axiom they are
m 1  m 4  [Fill in (1)]
and so
m 1  [Fill in (2)]
Statement/Reason: Make a similar statement/reason about
5 and 6 , ending on an expression
for 5 :
[Fill in (3)]
Statement/Reason: Since it is given that
with the above expressions, we get
1  5 , we have m 1  m 5 . By making substitutions
[Fill in (4)]
and a little algebra gives us that m 4  m 6 .
Statement/Reason: Since
4 and 6 are [Fill in (5)] (what kind of angles?) and
4  6 , we have
that l || m by the Parallelism in Absolute Geometry Theorem.
Proof of (2):
Given: Suppose that l and m lie in the plane and are cut by transversal t, forming the numbered
angles as shown above. Assume betweenness as is apparent. Suppose also that a pair of same side
interior angles are supplementary – say m 4  m 5  180 . [You take it from here and write the
proof…]
Q2: In the diagram below, find the values of x needed to guarantee that l m .
m ABH  2 x 50 , m FEG  150  3x
Then answer:

What theorem or axiom are you using that justifies your answer?
o
(1) If the lines are parallel, then alternate interior angles are congruent.
OR
o
(2) If alternate interior angles are congruent, then the lines are parallel.

Is this particular theorem or axiom equivalent to Euclid's parallel postulate, or is it a theorem
of absolute geometry (i.e. equivalent to the converse of the parallel postulate)?

Does this problem work in both hyperbolic and Euclidean geometries? Explain your answer.
Q3: In the diagram below, suppose that l m . What is the value of x?
m ABH  2 x 50 , m FEG  150  3x
(Why yes, it is the same diagram...and the same algebra. But not the same theorem!)
Then answer:

What theorem or axiom are you using that justifies your answer?
o
(1) If the lines are parallel, then alternate interior angles are congruent.
OR
o
(2) If alternate interior angles are congruent, then the lines are parallel.

Is this particular theorem or axiom equivalent to Euclid's parallel postulate, or is it a theorem
of absolute geometry (i.e. equivalent to the converse of the parallel postulate)?

Does this problem work in both hyperbolic and Euclidean geometries? Explain your answer.
Quadrilaterals in Euclidean geometry – the properties of parallelograms
Material from Kay, section 4.2
Theorem 1 on p. 212 states that
A diagonal of a parallelogram divides it into two congruent triangles.
DAC  BCA
There are several corollaries and related theorems on the properties of parallelograms. These are all
sketched out in the associated lecture (both literally sketched with pictures, and verbally talked
through on the accompanying sound buttons for each slide). I’m not going to make you write them
all out, since that’s just tedious – you should, of course, go through them all and be sure you’re
familiar with their proofs. It’s these, btw…
1.
2.
3.
4.
5.
6.
7.
8.
9.
Opposite sides of a parallelogram are congruent, as are opposite angles.
Adjacent angles are supplementary.
If a convex quadrilateral has opposite sides congruent, then it is a parallelogram.
If a convex quadrilateral has a pair of sides that are both congruent and parallel, then it
is a parallelogram.
The diagonals of a parallelogram bisect each other.
If the diagonals of a convex quadrilateral bisect each other, then the quadrilateral is a
parallelogram.
A parallelogram is a rhombus if and only if its diagonals are perpendicular.
A parallelogram is a rectangle if and only if its diagonals are congruent.
A parallelogram is a square if and only if its diagonals are both perpendicular and
congruent.
However, I would like you to write up a couple of them formally. Please write proofs for #7 and #8
on the list above.
Q4: Prove “A parallelogram is a rhombus if and only if its diagonals are perpendicular.”
Q5: Prove “A parallelogram is a rectangle if and only if its diagonals are congruent.”