Download Triangle Congruence, SAS, and Isosceles Triangles Recall the

Triangle Congruence, SAS, and Isosceles Triangles
Recall the definition of a triangle: 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
,
denoted by ªABC. The angles of ªABC are pA = pBAC, pB =
pABC, and pC = pACB.
There are other associated terms that we could formally define. These
include:
•
A side opposite an angle: In ªABC, side
and so on.
•
A vertex opposite a side: In ªABC, vertex B is opposite side
is opposite pC,
, and so on.
•
An angle included between two sides: In ªABC, pA is
included between
•
and
, etc.
A side included between two angles: In ªABC, side
included between pB and pC, etc.
is
Recall also the definition of congruence for segments and angles:
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
µpABC = µpXYZ.
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
:
,
:
,
:
pA : pX, pB : pY, pC : pZ.
, as well as
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.
We use this expression all the time as a shorthand way to express
the definition of congruence.
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:
1.
2.
3.
ªABC – ªABC
(Symmetric) If ªABC – ªXYZ, then ªXYZ – ªABC
(Transitive) If ªABC – ªXYZ, and ªXYZ – ªUVW, then
ªABC – ªUVW
(Reflexive)
These are not in your textbook, but they follow easily from the
definition, since congruence is based on equality of the real numbers
that are the measures of the segments and angles, and equality of real
numbers is reflexive, symmetric, and transitive.
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 Postulate
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). Then ªABC – ªXYZ.
Before we fully accept this as an axiom we will show that it is
independent of our current axiom set, by showing a model in which it
is true and one in which it is false.
The model in which it is true is regular Euclidean space as developed
in the Cartesian coordinate system. Since trigonometry holds in this
system, we can show that the third side of any triangle can be uniquely
determined if we know the opposite angle and the two other sides (by
using the Law of Cosines) and the other two angles will be uniquely
determined from the Law of Sines.
In the Taxicab geometry, in which angles are measured in the same
way as in the regular Euclidean model, SAS does not hold, as can be
demonstrated by these two triangles:
Definition: A triangle is said to be isosceles if it has a pair of
congruent sides. The angles opposite the congruent sides are the base
angles and the third angle is the apex of the triangle. The side between
the base angles is the base.
Theorem (The Isosceles Triangle Theorem): If a triangle is
isosceles, its base angles are congruent.
Two proofs:
~ Let ÎABC be isosceles with
we can find a point D interior to s
of
so
. By a previous theorem
uch that
. By crossbar, this ray intersects
is the bisector
at point E. Now
(given), and pACE – pBCE (bisector) and
and by CPCF,
. €
~ Let ÎABC be isosceles with
have
by SAS. Then
so the base angles are congruent. €
. Since
,
, we
by CPCF,
Yes, I know this seems like cheating, but it demonstrates the power of
the correspondence that is part of the definition of congruence.
This theorem is called the pos asinorum or “bridge of asses” because
of yet a third proof, the one used by Euclid. He extended the segments
to points D and E at equal distances from B and A,
respectively, then formed
. The result looks a little like
a bridge. It is also said that this proof was the bridge over which
students of mathematics had to pass if they were to be successful in
their study of geometry. It may be worth thinking
about how you would use this diagram in a proof.