Download =SAS Equal Side-Angle-Side Triangle Theorem

=SAS
Equal Side-Angle-Side Triangle Theorem
if p then q
Three equalities in the antecedent
Four equalities in the consequent
Antecedents (p)
“Two pairs of equal homologous legs and
a single pair of equal angles between the legs.”
(1) Side pair 1
(2) Angle pair
(3) Side pair 2
A
B
Consequents (q)
“Equal baselines, equal triangle areas,
and two pairs of equal homologous base
angles.”
(1) Baseline
(2) Triangle Area
(3) Base angle 1
(4) Base angle 2
B
D
C E
A
F
D
C E
F
Valid and Invalid
Triangle Congruency Theorems
To prove that triangles are congruent, one needs to know that at least three elements are equal.
Since there are three sides and three angles, that means there are six combinations of theorems.
SAS
ASA
SSS
AAS
AAA
ASS
Side-Angle-Side Valid Theorem
Angle-Side-Angle Valid Theorem
Side-Side-Side
Valid Theorem
Angle-Angle-Side Valid Theorem
Angle-Angle-Angle Invalid Theorem
Angle-Side-Side Invalid Theorem
(I.4)
(I.26, part 1)
(I.8)
(I.26, part 2)
?
The first four combinations form valid theorems, but the last two form invalid theorems. The four
valid theorems are represented below, but the proofs for these theorems will be shown elsewhere.
A
D
A
D
SAS
B
C
E
A
F
B
D
C E
A
F
D
ASA
B
C
E
A
F
B
D
C E
A
F
D
SSS
B
C
E
A
F
B
D
C E
A
F
D
SAA
B
C
E
F
B
C E
F
One can remember that the last two theorems are invalid by remembering this mnemonic: “If you
use an invalid theorem, you’re a first rate donkey, also known as a triple-A ass (AAA, ASS).”
A
A
D
D
AAA
B
C E
F
A
B
C E
A
D
F
D
ASS
B
C
E
F
B
C E
F
One can disprove each of these theorems by a method called “counter-example”. What we need
to do for the first is show that two triangles exist which have all homologous angles equal
(AAA), but which are not congruent. For the second, we need to show that two triangles exist
which have one pair of equal homologous base angles and two pairs of equal homologous legs
(ASS), but which are not congruent.
The first counter example is pretty easy to make: simply cut the leg of any triangle ABC with a
point E and draw a straight line through E parallel to the base BC. The triangle AEF is a part of
the whole triangle, and since the bases are parallel, the homologous base angles form equal
corresponding angles. The two triangle have all homologous angles equal, but they are not
congruent.
A
AAA
counterexample
E
F
B
A
A
C
E
B
C
F
The second counter example is pretty easy to make: after using a circle to make a triangle ABC
with equal legs AB and AC, extend the base BC any random distance to D, then draw a straight
line AD from the end of the extension to the vertex of the triangle. In triangles ACD and ABD,
the base angle at D and the leg AD are common, while the homologous legs AC and AB are also
equal, so the two triangle have one pair of equal homologous base angles and two pairs of equal
homologous legs (ASS); but the whole triangle is not congruent with the part.
ASS
counterexample
A
D
C
A
D
C
B
A
D
B