44 Distance; degree-measure of an angle Ch. 3
3.6 Mid-line of an angle-support
3.6.1 Right-angles
Definition. Given any pointP=Aof a lineAB,byA 5 (v) there is a half-line[P,Q
such that|∠APQ|◦=90. Then∠APQis called aright-angle.
IfR=Pis such thatP∈[A,R]then∠RPQis also a right-angle. For|APRis a
straight angle-support, so having supplementary angles,
|∠APQ|◦+|∠QPR|◦= 180.As|∠APQ|◦=90 it follows that|∠RPQ|◦= 180 − 90 =90.
3.6.2 Perpendicularlines
Definition.Ifl,mare lines inΛ, we say thatlisperpendiculartom, writtenl⊥m,
iflmeetsmat some pointPand ifA=Pis onl,andQ=Pis onm,then∠APQis a
right-angle.
COMMENT. In 3.6.1, we say that aperpendicularPQhas been erectedto the
lineABat the pointPon it.
Perpendicularity has the following properties:-(i)If l⊥m, then m⊥l.(ii)If l⊥m, then l=m and l∩m=0./Proof.
These follow immediately from the definition of perpendicularity.P
A
Q
90
Figure 3.10. Perpendicular lines.
A B
C P
P′
Mid-line of an angle-support.