A Summary of Triangle Congruence
Definition of Triangle Congruence
We say that triangle ABC is congruent to triangle DEF if
- AB = DE
- BC = EF
- CA = FD
- Angle A = Angle D
- Angle B = Angle E
- Angle C = Angle F
(Of course Angle A is short for angle BAC, etc.)
Very Important Remark about Notation (ORDER IS CRITICAL):
Notice that saying triangle ABC is congruent to triangle DEF is not the same as saying triangle ABC is congruent to triangle FED. For example the first statement means, among other things, that AB = DE and angle A = angle D. The second statement says that AB = FE and angle A = angle F. This is very different!
The notation convention for congruence subtly includes information about which vertices correspond. To write a correct congruence statement, the implied order must be the correct one.
The good feature of this convention is that if you tell me that triangle XYZ is congruent to triangle CBA, I know from the notation convention that XY = CB, angle X = angle C, etc. So once the order is set up properly at the beginning, it is easy to read off all 6 congruences.
Congruence Criteria
It turns out that knowing some of the six congruences of corresponding sides and angles are enough to guarantee congruence of the triangle and the truth of all six congruences.
Side-Angle-Side (SAS)
This criterion for triangle congruence is one of our axioms. So we do not prove it but use it to prove other criteria.
Using words:
If two sides in one triangle are congruent to two sides of a second triangle, and also if the included angles are congruent, then the triangles are congruent.
Using labels:
If in triangles ABC and DEF, AB = DE, AC = DF, and angle A = angle D, then triangle ABC is congruent to triangle DEF.
Side- Side-Side (SSS)
Using words:
If 3 sides in one triangle are congruent to 3 sides of a second triangle, then the triangles are congruent.
Using labels:
If in triangles ABC and DEF, AB = DE, BC = EF, and CA = FD, then triangle ABC is congruent to triangle DEF.
Proof: This was proved by using SAS to make "copies" of the two triangles side by side so that together they form a kite, including a diagonal. Then using what was proved about kites, diagonal cuts the kite into two congruent triangles.
Details of this proof are at this link. The similarity version of this proof is B&B Principle 8.
Angle-Side-Angle (ASA)
Using words:
If two angle in one triangle are congruent to two angles of a second triangle, and also if the included sides are congruent, then the triangles are congruent.
Using labels:
If in triangles ABC and DEF, angle A = angle D, angle B = angle E, and AB = DE, then triangle ABC is congruent to triangle DEF.
Proof: This proof was left to reading and was not presented in class. Again, one can make congruent copies of each triangle so that the copies share a side. Then one can see that AC must = DF.
For the proof, see this link. The similarity version of this proof is B&B Principle 6.
Side-Side-Angle (SSA) not valid in general
Using labels:
SSA would mean for example, that in triangles ABC and DEF, angle A = angle D, AB = DE, and BC = EF.
With these assumptions it is not true that triangle ABC is congruent to triangle DEF. In general there are two sets of congruent triangles with the same SSA data.
Examples were investigated in class by a construction experiment. There is also a Java Sketchpad page that shows why SSA does not work in general.
Hypotenuse-Leg (HL) for Right Triangles
There is one case where SSA is valid, and that is when the angles are right angles.
Using words:
In words, if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the triangles are congruent.
Using labels
If in triangles ABC and DEF, angle A = angle D = right angle, AB = DE (leg), and BC = EF (hypotenuse), then triangle ABC is congruent to triangle DEF.
Proof:
The proof of this case again starts by making congruent copies of the triangles side by side so that the congruent legs are shared. The resulting figure is an isosceles triangle with altitude, so the two triangles are congruent.
For the details of the proof, see this link. The similarity version of this theorem is B&B Corollary 12a (the B&B proof uses the Pythagorean Theorem, so the proof is quite different).
