Sec. 4.3 Properties of triangles and half-planes 55
Proof.
(i) and (ii) follow immediately from the definition, while (iii) follows from 4.2.1.
(iv) As perpendicular lines form right-angles with each other at some point,l
must meetnat some pointA,andmmust meetnat some pointPsuch that ifB
is any other point oflandQis any point ofmon the other side ofnfromB,then
|∠PAB|◦= 90 ,|∠APQ|◦=90. Then, as these are alternate angles equal in measure,
by 4.2.1 Corollary 1,l‖m.
4.3 Properties of triangles and half-planes
4.3.1 Side-angle relationships; the triangle inequality
If A,B,C are non-collinear points and|A,B|>|B,C|,then|∠ACB|◦>|∠BAC|◦,so
that in a triangle a greater angle is opposite a longer side.
A
C
B
D
Figure 4.7. Angle opposite
longer side.A
B
C
D
Figure 4.8. The triangle inequality.Proof. ChooseD∈[B,Aso that|B,D|=|B,C|.ThenD∈[B,A]as|B,D|<|B,A|.
Now|∠ACB|◦>|∠DCB|◦as[C,D ⊂IR(|BCA),and|∠DCB|◦=|∠BDC|◦by
4.1.1. But|∠BDC|◦>|∠DAC|◦by 4.2.1 Corollary 1, so
|∠ACB|◦>|∠DCB|◦=|∠BDC|◦>|∠DAC|◦.Hence|∠ACB|◦>|∠DAC|◦and∠DAC=∠BACasD∈[B,A].
COROLLARY.If A,B,C are non-collinear points and|∠ACB|◦>|∠BAC|◦,then
|A,B|>|B,C|, so that in a triangle a longer side is opposite a greater angle.
Proof.Forif|A,B|≤|B,C|,wehave|∠ACB|◦≤|∠BAC|◦by 4.1.1(i) and this
result.
The triangle inequality.If A,B,C are non-collinear points, then|C,A|<
|A,B|+|B,C|.
Proof. Take a pointDso thatB∈[A,D]and|B,D|=|B,C|.
As[C,B ⊂IR(|ACD)we have|∠DCA|◦>|∠DCB|◦.But|∠DCB|◦=|∠CDB|◦