260 CHAPTER 8 GEOMETRY FOR AMERICANS
but in practice we omit the f':.. symbol unless the context is ambiguous.Fact 8.2. 1 Congruence Conditions. If you travel around a triangle, say, counterclock
wise, starting at an angle, you will encounter, in order, a side, then an angle, etc.
Suppose that you travel around a triangle and record the length of a side, the measure
of the angle, then of the next side. Then suppose you do the same thing with another
triangle, and the values recorded are the same. Then the two triangles are congruent.This condition is often abbreviated as SAS. There are several congruence conditions:
SAS, ASA, SSS, AAS.
Fact 8.2.2 Note that ASS does not guarantee congruence (give a counterexample),
unless the angle is greater than or equal to 90° or the second side is longer than the first.
And of course AAA does not mean congruence; it indicates similarity (see page 274).Fact 8.2.3 The Triangle Inequality. In a triangle, the sum of the lengths of any two
sides is strictly greater than the third side.Fact 8.2.4 Angle Inequalities in Triangles. Let ABC be a triangle. If mLA > mLB,
then BC > AC, and conversely.
8.2.5 Vertical Angles. When two lines meet in a point, four angles are formed in
two pairs of opposite angles. These opposite angles are also called vertical angles.Vertical angles are equal. For example, LECD = LACB in the figure below.
(Remember, the absence of the label "Fact" means that this is an exercise for you to
prove. It should be rather easy.)Parallel Lines
In the figure below, lines AB and CD are parallel, i.e. AB II CD. The line BC that
intersects the two parallel lines is called a transversal.AD