Definition: A triangle is the union of three
segments (called its sides), whose endpoints
(called its vertices) are taken, in pairs, from a
set of three noncollinear points. Thus, if the
vertices of a triangle are A, B and C, then its
sides are
,
, and
, and the triangle is
then the set defined by
c
c
,
denoted by ªABC. The angles of ªABC are
pA = pBAC, pB = pABC, and pC = pACB.
There are another of associated terms that we
could formally define. These include:
A side opposite an angle
A vertex opposite a side
An angle included between two sides
A side included between two angles
Definition: Two segments
and
are said
to be congruent (we write
–
) if and
only if AB = XY. Two angles pABC and
pXYZ are said to be congruent (and we write
pABC – pXYZ) if and only if mpABC =
mpXYZ.
Definition: A correspondence between vertices
of two triangles is a one-to-one, onto mapping
from the set of vertices of the first triangle to the
second.
Intuitively, a correspondence is a matching of
vertices between two triangles. This matching
also establishes a correspondence between sides
and angles.
Shorthand: ABC : XYZ iff A : X, B : Y, and
C : Z. This also establishes
:
,
: ,
:
, as well as
pA : pX, pB : pY, pC : pZ.
Definition: If, under a certain correspondence
between the vertices of two triangles,
corresponding sides and corresponding angles
are congruent, the triangles are said to be
congruent.
Recall that a definition is really an “if and only
if” statement. We could reword this as
Two triangles are said to be congruent if and
only if there is a correspondence between the
vertices such that corresponding sides and
corresponding angles are congruent.
We abbreviate this definition by saying,
“Corresponding parts of congruent figures are
congruent,” and further abbreviate this to CPCF.
Notation: If ªABC is congruent to ªXYZ, we
write
ªABC – ªXYZ.
Note: This notation not only says that the two
triangles ªABC and ªXYZ are congruent, but
also establishes the correspondence under
which they are congruent. (The
correspondence is ABC : XYZ).
Thus,
ªABC – ªXYZ
if, and only if,
–
,
– ,
–
pA – pX, pB – pY, pC – pZ
Properties of Congruence for Triangles:
ªABC – ªABC
(Symmetric) If ªABC – ªXYZ, then
ªXYZ – ªABC
(Transitive) If ªABC – ªXYZ, and
ªXYZ – ªUVW, then ªABC – ªUVW
1. (Reflexive)
2.
3.
Note: Because the statement ªABC – ªXYZ is
a statement about a particular correspondence,
the statements ªABC – ªCBA says something
very different from ªABC – ªABC.
Among other things, ªABC – ªCBA implies
that pA–pC, which is not implied by ªABC –
ªABC
The SAS Hypothesis:
Under the correspondence ABC : XYZ, let two
sides and the included angle of ªABC be
congruent, respectively, to the corresponding
two sides and the included angle of ªXYZ (that
is, for example,
–
,
– , and
pA – pX).
If this hypothesis is satisfied, we want to be able
to conclude
ªABC – ªXYZ
We cannot do it from the current axioms. We
will eventually adopt this as our penultimate
axiom. First, however, we show that it is
independent of our current axiom set.