Download Sect. 6.4 SSA

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.
C
C
C
B?
A
A
B
A
B1
B2
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.
Z
C
A
B
X
Y
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.
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.
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.
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.
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.
Corollary E: If in a triangle a median bisects the vertex angle, the triangle is isosceles.
~ Apply SSA theorem and reach a contradiction to the Exterior Angle Inequality. 
Corollary F (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.
Z
C
A
B
X
Y