Geometry with Trigonometry

18 Preliminaries Ch. 1

then[A,B,C]is congruent to[A′,B′,C′]. This is known as the ASA (angle, side, angle)
condition for congruence of triangles.
It can be proved that ifTandT′are triangles with vertices{A,B,C},
{A′,B′,C′}, respectively, for which

|B,C|=|B′,C′|,|C,A|=|C′,A′|,|A,B|=|A′,B′|,

thenTis congruent toT′. This is known as the SSS(side-side-side) principle of con-
gruence.

1.5.5 Parallellines..............................

1. Distinct linesl,mare said to beparallelifl∩m=0; this is written as/ l‖m.We
   also takel‖l.

l

m

Figure 1.20. Parallel lines.

By observation, given any lineland any pointPthere cannot be more than one
linemthroughPwhich is parallel tol.

A

B

P

R

Q

◦

◦

Figure 1.21. Alternate angles for a transversal.

A

B

P

R

Q

S

T

◦

◦

Corresponding angles.

It can be shown that two lines are parallel if and only ifalternateangles made
by a transversal, as indicated, are equal in magnitude, or equivalently, if and only if
correspondingangles made by a transversal are equal in magnitude.

2. A convex quadrilateral in which opposite side-lines are parallel to each other is
   called aparallelogram. A parallelogram in which adjacent side-lines are perpendic-
   ular to each other is called arectangle.