l’
are two lines cut by a transversal
t
in such a way that a pair of
corresponding angles are congruent, then the two lines are parallel.
Corollary 4.4.5
If
l
and
l’
are two lines cut by a transversal
t
in such a way that two
nonalternating interior angles on the same side of
t
are supplements, then
the two lines are parallel.
Corollary 4.4.6
If
l
is a line and
P
is an external point, then there is a line
m
such that
P
lies on
m
and
m
is parallel to
l
.
Proof
is mainly a famous construction “the double perpendicular construction of a parallel
line”:
Drop one perpendicular from
P
to
l
and call the point at the foot of the perpendicular Q.
Construct
QP
uur
s
.
Next construct a line perpendicular to
QP
uur
s
through
P
.
(this is the second perpendicular in the construction, hence “double”).
This is NOT the proof, though.
That’s in the book.
Please read it and learn it.
Corollary 4.4.7
The Elliptic Parallel Postulate is false in any model of neutral geometry.
Corollary 4.4.8
If
l, m
, and
n
are any three lines such that
m
l
⊥
and
n
l
⊥
then either
m = n
or
m
is parallel to
n
.
16
4.5
The SaccheriLegendre Theorem
Let A, B, and C be three noncollinear points.
The angle sum,
σ,
for
ABC
∆
is the sum of the
measures of the three interior angles of this triangle.
Theorem 4.5.2
SaccheriLegendre Theorem
If
ABC
∆
is any triangle then
σ(
ABC
∆
)
≤
180°.
Note:
NOT equal.
This covers the Hyperbolic case as well as the Euclidean case.
Lemma 4.5.3
If
ABC
∆
is any triangle, then
(
)
(
)
180
CAB
ABC
μ
μ
∠
+
∠
<
°
Let
ABC
∆
be any triangle
Let D be a point on
AB
uuur
such that A*B*D.
(
)
(
)
180
CBD
ABC
μ
μ
∠
+
∠
=
°
because they
are a linear pair.
Further
CBD
∠
is an exterior angle to the
triangle so
(
)
(
)
CAB
CBD
μ
μ
∠
<
∠
.
Substituted in the equation above, we find
(
)
(
)
180
CAB
ABC
μ
μ
∠
+
∠
<
°
. □
Lemma 4.5.4
If
ABC
∆
is any triangle and
E
is a point on the interior of side
BC
, then
(
(
)
(
)
180
ABE
ECA
ABC
σ
σ
σ
∆
+
=
+
°
.
Assume the hypothesis.
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
ABE
ECA
EAB
ABE
BEA
CAE
ECA
AEC
σ
σ
μ
μ
μ
μ
μ
μ
∆
+
∆
=
∠
+
∠
+
∠
+
∠
+
∠
+
∠
By the Angle Addition Postulate (since
E
is
in the interior of
ACB
∠
, we know that
(
)
(
)
(
)
EAB
CAE
BAC
μ
μ
μ
∠
+
∠
=
∠
17
A
B
C
D
A
B
C
E
Now
(
)
(
)
180
BEA
AEC
μ
μ
∠
+
∠
=
°
because they
are a linear pair.
Combining these facts we get
(
)
(
)
(
)
(
)
(
)
180
(
)
180
ABE
ECA
BAC
ABE
ECA
ABC
σ
σ
μ
μ
μ
σ
∆
+
∆
=
∠
+
∠
+
∠
+
° =
∆
+
°
□
Lemma 4.5.5
If
A, B,
and
C
are three noncollinear points, then there exists a point
D
that does not lie on
AB
uur
s
such that
(
)
(
)
ABD
ABC
σ
σ
∆
=
∆
and the angle
measure of one of the interior angles in
ABD
∆
is less than or equal to
0.5
(
)
CAB
μ
∠
.
Let
A, B,
and
C
be three noncollinear points.
Construct
ABC
∆
.
Let
E
be the midpoint of
BC
.
Let D be a point on
AE
uuur
such that A*E*D and AE = ED.
Why is
AEC
DEB
∆
2245 ∆
?
Now since they are congruent, the angle sums will be equal.
Now, why do we know that:
(
)
(
)
(
)
180
ABC
ABE
AEC
σ
σ
σ
∆
=
∆
+
∆

°
?
By similar reasoning
(
)
(
)
(
)
180
ABD
ABE
DEB
σ
σ
σ
∆
=
+
∆

°
So how do we know that
18
A
B
C
E
E
A
B
C
D
(
)
(
)
ABD
BAC
σ
σ
∆
=
∆
?
This completes half of the proof.