Wednesday, March 22, 2006

Congruent Triangles

In today's blog, I review congruent triangles. These are all based on Euclid's Elements. If you would like to review Euclid's classic works, I strongly recommend David Joyce's web site on Euclid's Elements.

The site presents the complete Elements with proofs, definitions, and axioms and includes commentaries. All of the applets on this page are taken from David's web site.

Definition 1: Congruent Triangles

Two triangles are said to be congruent if corresponding angles are congruent and corresponding sides are congruent.

Postulate 1: Congruent by Side-Angle-Side (SAS)

For any two triangles, if two corresponding sides are congruent and the angle in between is congruent, then the triangles are congruent.

In the example below, if AB ≅ DE and BC ≅ EF and ∠ ABC ≅ ∠ DEF; then triangle ABC ≅ triangle DEF.

Euclid originally presented this as a proof using the method of superposition in his Elements (see here). The rigorousness of this method has been questioned. For these reasons, I am presenting this as a postulate. For those interested in more details, see here.

Postulate 2: Congruent by Side-Side-Side (SSS)

For any two triangles, if all 3 corresponding sides are congruent, then the triangles are congruent.

Euclid provides a proof of this one also using the method of superposition. For my purposes, I am presenting it as a postulate. For Euclid's proof, see here.

Lemma 1: For any two triangles, if two corresponding angles are congruent and the sides in between those angles are congruent, then the triangles are congruent.
If ∠ ABC ≅ ∠ DEF and BC ≅ EF and ∠ BCA ≅ EFD, then triangle ABC ≅ triangle DEF

Proof:

(1) Assume that AB is not congruent to DE

(2) Since they are not congruent, either AB or DE is bigger. We will assume it is AB (if it were DE we could still use the same argument)

(3) Then there exists a point G on AB such that BG ≅ DE

(4) By our assumption, we have ∠ ABC ≅ ∠ DEF and we have BC ≅ EF

(5) So, then we have triangle GBC ≅ triangle DEF [By Postulate 1 above]

(6) But then ∠ BCG ≅ ∠ EFD since corresponding angles of congruent triangles are congruent.

(7) But this is impossible since ∠ BCA ≅ &EFD and ∠ BCA is clearly not equal to ∠ BCG.

(8) Since we have a contradiction we reject our assumption at (1) and conclude that AB ≅ DE

(9) But now we have triangle ABC ≅ triangle DEF by Postulate 1 above since:

AB ≅ DE [By Step #8]

∠ ABC ≅ ∠ DEF [By the given]

BC ≅ EF [By the given]

QED

Corollary 1.1: For two triangles, if any two angles are congruent and any side is congruent, then both triangles are congruent.

(1) Let us assume that we have the pattern AAS (angle-angle-side) or SAA (side-angle-angle) congruent for the two triangles.

If we have ASA, then we know they are congruent by Lemma 1.

(2) But then we know that all corresponding angles of both triangles are congruent since:

Let a,b,c be the three angles of the first triangle. Let a',b',c' be the three angles for the second triangle.

Let's suppose that a,b match so that a' = a, b' = b.

Now, we know that a+b+c = 180 and a' + b' + c' = 180 [See Lemma 4 here]

So c = 180 - a - b

Also c' = 180 - a' - b' = 180 - a - b

So we see that c = c'

(3) And (#2) means that AAS and SAA imply ASA (angle-side-angle) which implies congruence by Lemma 1 above.

QED

References