Download A Digression into SSA or, as the Textbook Prefers, ASS Recall the

A Digression into SSA or, as the Textbook Prefers, ASS
Recall the “ambiguous case” from trigonometry: Here, you were asked
to “solve” a triangle given two sides and an angle opposite one side.
Before applying the law of sines, you needed to check to see how many
triangles you had – none, one, or two. Thus the information you were
given did not uniquely determine a single triangle.
For example, referring to the diagram below, pCAB –pZXY, AC =
XZ, and CB = ZY, but clearly ªABC is not congruent to ªXYZ. Thus
the congruence of two sides and a non-included angle of one triangle
to the corresponding two sides and non-included angle of another
triangle is not enough to guarantee congruence of the triangles.
However, we can 1) say some things about the relation between pB
and pY, and 2) make some restrictions that will guarantee congruence
in some cases.
Theorem (SSA Theorem - Not in the Text): If, under some
correspondence between their vertices, two triangles have two pairs of
corresponding sides and a pair of corresponding angles congruent, and
if the triangles are not congruent under this correspondence, then the
remaining pair of angles not included by the congruent sides are
supplementary.
~
The proof reduces to SAS if the angle is included between the two
sides, so assume otherwise. Given ªABC and ªXYZ with pCAB
–pZXY, AC = XZ, and CB = ZY, but ªABC á ªXYZ; we show
pABC and pXYZ are supplementary.
If pACB –pXZY, then the triangles are congruent by ASA. So,
WLOG, µ(pACB) > µ(pXZY). Find ray
and with µ(pACP) = µ(pXZY).
between
intersects
and
at some point D
with A*D*B. By ASA, ªADC / ªXYZ. By CPCF, pADC / pXYZ
and CD = ZY. Since pADC and pCDB are a linear pair, they are
supplementary, so µ(pADC) + µ(pCDB) = 180, so µ(pXYZ) +
µ(pCDB) = 180. Finally, since CB = ZY = CD, ªCDB is isosceles so
µ(pABC) = µ(pCDB), so we have µ(pXYZ) + µ(pABC) =180. €
Some Easy Corollaries:
Corollary A: If, under some correspondence between their vertices,
two acute triangles have two sides and an angle opposite one of them
congruent, respectively, to the corresponding two sides and angle of
the other, the triangles are congruent.
~ The hypothesis of this theorem satisfies that of the SSA theorem, so
if the triangles are not congruent, the two remaining angles are
supplementary. But this cannot be if all angles in both triangles are
acute. So the triangles must be congruent. €
Corollary B (HL Theorem): If the hypotenuse and leg of one right
triangle are congruent, respectively, to the hypotenuse and leg of a
second right triangle, the two triangles are congruent.
~ Since the hypothesis of this theorem satisfies that of the SSA
theorem, if the two triangles are not congruent, the remaining (nonright) angles must be supplementary. However, in a right triangle,
these angles must be acute, and so cannot be supplementary. Thus the
triangles are congruent. €
Corollary C (HA Theorem): If the hypotenuse and one acute angle
of one right triangle are congruent, respectively, to the hypotenuse and
acute angle of a second right triangle, the two triangles are congruent.
~ Follows immediately from AAS Theorem. €
Corollary D (LA Theorem): If a leg and one acute angles of one
right triangle are congruent, respectively, to a leg and acute angle of a
second right triangle, the two triangles are congruent.
~ Follows immediately from AAS Theorem. €
Cororllary E (SsA Congruence Theorem – Not in the Text): Given
ªABC and ªXYZ, suppose pCAB –pZXY, AC = XZ, CB = ZY, and
CB > CA. Then the two triangles are congruent.
~ If pCAB –pZXY, AC = XZ, CB = ZY, then by the SSA Theorem,
if the triangles are not congruent, pB and pY are supplementary.
Thus, either they are both right angles or one is obtuse. In either case,
they must be opposite the longest side of the triangle. But ZY = CB >
CA = XZ, a contradiction. Thus the triangles must be congruent. €
Intuitively, this is the case where
is longer than
and so cannot
“pivot around” the point C to come to rest on two different places on
the side opposite C.