Triangle Congruence

What congruent means

Two figures are congruent if they have the same size and the same shape. Put another way, you can move one onto the other using only rigid motions: slides (translations), turns (rotations) and flips (reflections). A rigid motion never stretches or bends anything, so every length and every angle stays the same.

A triangle has six parts: three sides and three angles. If two triangles are congruent, all six pairs of matching parts are equal. The good news of this unit is the other direction: to show two triangles are congruent, three well-chosen pairs are enough.

Matching parts, naming order and marks

The statement △ABC ≅ △DEF is read “triangle ABC is congruent to triangle DEF.” The order of the letters is the important part. Letters in the same position match:

• A ↔ D, B ↔ E, C ↔ F, so ∠A = ∠D, ∠B = ∠E, ∠C = ∠F.
• Sides use the same positions: AB = DE, BC = EF, CA = FD.

So △ABC ≅ △DEF is a different claim from △ABC ≅ △EDF. You may write the same fact as △BCA ≅ △EFD, as long as both names move together.

On diagrams, tick marks show equal sides (one tick matches one tick, two match two) and arcs show equal angles. A small square marks a right angle.

The five shortcuts

AAS works because the third angles must match too (angles add to 180°), which turns it into ASA. HL works because the Pythagorean theorem gives the third sides, so it becomes SSS.

Why SSA and AAA don’t work

SSA (two sides and an angle that is not between them) can fit two different triangles. Fix angle A and side AB. The third side starts at B and has a fixed length, so its end lies on a circle around B. That circle can cross the ray from A twice, at C₁ and C₂. Both triangles have the same marked parts, but they are not congruent. (When the side opposite the angle is at least as long as the other given side, there is only one triangle. That is why the right-angle case, HL, works: the hypotenuse is always the longest side.)

AAA (three angles) fixes the shape but not the size. A photo and its enlargement have the same angles. Such triangles are similar, the topic of the next unit, but not congruent.

CPCTC

CPCTC stands for “corresponding parts of congruent triangles are congruent.” Once a shortcut proves two triangles congruent, every other pair of matching parts is equal too. This is how most proofs end: you prove the triangles congruent first, then use CPCTC to get the side or angle you actually wanted.

Two ways to write a proof

Given: M is the midpoint of AC and of BD. Prove: AB = CD.

Notice the order in step 4: A ↔ C, M ↔ M, B ↔ D. That is what makes AB match CD in step 5.

The same proof as a paragraph: Since M is the midpoint of both segments, AM = CM and BM = DM. Angles AMB and CMD are vertical angles, so they are equal. That gives two sides and the included angle, so △AMB ≅ △CMD by SAS, and AB = CD because corresponding parts of congruent triangles are congruent.

The isosceles triangle theorem

Theorem: if two sides of a triangle are equal, the angles opposite them (the base angles) are equal.

Proof: Let M be the midpoint of BC and draw AM. Then AB = AC (given), BM = CM (M is the midpoint) and AM = AM (the same segment). So △ABM ≅ △ACM by SSS, and ∠B = ∠C by CPCTC.

This is Euclid’s Book I, Proposition 5. His own proof is longer and uses SAS on a pair of overlapping triangles. It has been called the pons asinorum, Latin for “bridge of asses.” Joyce notes that it isn’t clear whether the name comes from how hard students found it or from the bridge-like shape of Euclid’s figure.

Real-world uses: why triangles are rigid

SSS has a practical meaning. If three bars of fixed length are pinned together at their ends, the triangle they make has only one possible shape. A four-sided frame does not have this property: with the same four side lengths it can lean into many shapes.

That is why you see triangles in bridge trusses, roof frames, bike frames, cranes and the diagonal braces on gates and shelves. Adding one diagonal to a rectangle splits it into two triangles and stops it from leaning.

Common mistakes

A short history: Euclid’s three congruence propositions

Euclid’s Elements, written around 300 BCE, proves the main shortcuts in Book I:

• Proposition 4 is SAS. Euclid proves it by placing one triangle on top of the other (superposition). Joyce notes that the modern approach of David Hilbert takes part of SAS as an axiom instead.
• Proposition 8 is SSS.
• Proposition 26 covers both ASA and AAS.

Euclid did not use the word “congruent” or names like SAS; Joyce points out that calling these “congruence theorems” is a later label. Euclid gives no SSA proposition, and Joyce notes that the right-angle case (HL) is proved twice inside the proof of Book III, Proposition 14.

You’ll get three kinds: pick the shortcut (or “Not enough info”) from a marked pair, name matching parts, and solve for x. For parts, type letters like EF or F; for numbers, type whole numbers.

Sources

• Euclid, Elements, Book I, ed. D. E. Joyce, Clark University: Proposition 4 (SAS, with Joyce’s note on Hilbert); Proposition 5 (isosceles base angles, pons asinorum); Proposition 8 (SSS); Proposition 26 (ASA and AAS, with Joyce’s notes on SSA and the right-angle case).
• MacTutor History of Mathematics, University of St Andrews: Euclid of Alexandria (biography).

Credits: all diagrams on this page are original SVG drawings made for Jugami. The isosceles proof here uses SSS with a midpoint, not Euclid’s own construction. Quotations and notes on the Elements follow Joyce’s site.