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Chapter 2
One of the basic axioms of Euclidean geometry says that two points
determine a unique line.
EXISTENCE AND UNIQUENESS
Two lines that don’t intersect are called parallel.
This implies that two distinct lines cannot intersect in two or more points, they can either intersect in only one point or not at all.
PROBLEMGiven a line and a point P not on ‚
construct a line through P and parallel to.
lll
L
let A be any point on , and draw Then draw a line so that as shown in the figure.
This will be the desired line .
AP
PABQPA PQl
If and are not parallel, we may assume without loss of generality that they intersect
as in the figure on the side of B at the point C.
PQ l
Now consider . the exterior angle is equal to the interior angle . But this contradicts the exterior angle theorem, which
states that .
PAC APQPAC
PACQPA PQ l
The proof will be by contradiction .
Hence must be parallel to.
I
Given a line and a point P not on,
there exists a line that contains P
and is parallel to.
l l
l
COROLLARY
Given lines AB PQ
,then is parallel PABQPA AB
PQto .
if
as in the figure,and
THE PARALLEL POSTULATE
If is any line and P is a point not on . l
l
l
parallel to.
then there is no more than one line through P
Opposite Interior Angles Theorem
are opposite interior angles.
Let and be parallel lines withPQ
PABQPA
transversal such that and
QPA PABAP
AB
Then .
The proof will be by contradiction.
then we could
If the theorem
and if PABQPA construct a distinct line QP
PABPABQAP through P such that
. QAP are opposite interior angles, their congruence
implies that ABQP .
Since and
was false ,
and are two different lines , each goesQP PQ
But this is now a contradiction of the parallel postulate :
So these angles must be congruent.
we assumed that
through P and each is parallel to .
This contradiction comes about because
AB
PABAPQ .
THEOREM
Let be any triangle. thenABC
180 CBA .
ProofLet be the line through A parallel to
such that
angles and and are opposite
interior angles, as in figure. so
Hence
PAQ BC
BAPB
B BAPC CAQ
CAQBAPACBA
and are opposite interior
CAQC and .
=180.Since , and CAQBAPA all together
make a straight line.
Let be any quadrilateral. then
360 DCBA
ABCD
We draw the diagonal AC thus breaking the
quadrilateral into two triangles. Note that
DCBA
DACDACBBCADCAB
)( DACDACD ).( BCABCAB
+
The first sum of the last expression represents
the sum of the angles of ACDand the second sum represents the sum of the
angles of CAB . Hence,
each is 180 and together they add up to 360.
COROLLARY(SAA)
In and assume that
EFBC ABC DEF
EBDA , and
DEFABC then
.
.
Given a quadrilateral ABCD, the following are equivalent :
AB.1 AD BCCD
CDAB .2 BCAD
and
and
.
.
3.The diagonals bisect each other.
LEMMA
Let be a line. P a point not on. l lAnd A and B distinct points on lsuch that
PAis perpendicular to l. Then PBPA .
L
THEOREM
Let and be parallel lines and let P and Q
Then the distance from P to
2l1lbe points on 2lequals the distance from Q to
1l
1l
Proof
Draw lines from P and from Q perpendicular
to1l, Meeting
1l at B and at C , respectively.
90PBCSince 90QCBand ,
these angles are congruent . , Moreover
is congruent to the supplement of QBC PBC .
2l
1l
So By opposite interior angles.
must be a parallelogram, since opposite sides are Parallel . Hence
PB
PQ
QC
BCSimilarly, . Therefore PBCQ
QCPB ,as claimed .