=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
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
