Geometry with Trigonometry

54 Congruence of triangles; parallel lines Ch. 4

COMMENT.IfD and
H are on opposite sides
of BC,then∠CBH and
∠BCDare known asal-
ternate angles. This last
result implies that if al-
ternate angles∠CBHand
∠BCDare equal in mea-
sure, then CD and BH
cannot meet at some point
A.

A

B

D

C

E

G

H

F

Figure 4.5. Result on alternate angles.
COROLLARY 2.Given any line l and any point P∈l, there is a line m which
contains P and is such that l∩m=0./
Proof. Take any pointsA,B∈land lay off an angle∠APQon the opposite side
ofAPfromB,sothat|∠APQ|◦=|∠PAB|◦. Then by the recent comment the line
PQdoes not meetl.Inthis∠APQand∠PABare alternate angles which are equal in
measure.

4.2.2 Parallellines..............................

Definition.Iflandmare lines inΛ, we say thatlisparalleltom, writtenl‖m,if
l=morl∩m=0./

Parallelism has the following properties:-

(i)l‖l for all l∈Λ;

(ii) If l‖mthenm‖l;

(iii)Given any line l∈Λand any point P∈Π, there is at least one line m which

contains P and is such that l‖m.

(iv) If the lines l and m are both perpendicular to the line n, then l and m are

parallel to each other.

A

B

Q

P

l

m

Figure 4.6. Parallel lines.

A

B

Q P

l

n m