Congruent Figures

Part 8, Chapter 8: Plane Isometries, Rigid Motions

Learning objectives

Define congruence of plane figures via the existence of an isometry mapping one to the other
Recognise that congruence is an equivalence relation (reflexive, symmetric, transitive)
Apply the SSS, SAS, and ASA criteria for triangle congruence
Explain why SSA is not a valid congruence criterion (the ambiguous case)

Congruence is just “moveable equality.” Two figures are congruent if you can carry one onto the other by some rigid motion of the plane. With four kinds of isometry available, this gives a precise meaning to a notion you have used since elementary school, and the meaning turns out to have the algebraic shape of equality itself.

The definition

Two plane figures $F_1$ and $F_2$ are congruent, written $F_1 \cong F_2$ , if there exists an isometry $f$ with $f(F_1) = F_2$ .

This compresses the old “same shape and size” intuition into one mathematical sentence: the figures are not literally identical (they may sit in different places), but a rigid motion explains the difference completely.

Congruence is an equivalence relation

The relation $\cong$ obeys the three axioms that make it behave like equality:

Reflexive. $F \cong F$ . Take $f = \mathrm{id}$ .

Symmetric. If $F_1 \cong F_2$ via isometry $f$ , then $F_2 \cong F_1$ via $f^{-1}$ .

Transitive. If $F_1 \cong F_2$ via $f$ and $F_2 \cong F_3$ via $g$ , then $F_1 \cong F_3$ via $g \circ f$ .

The three axioms above are exactly the closure properties of the isometry group (identity, inverse, composition) recast as facts about figures.

How many isometries are enough?

A useful technical fact: any two congruent figures in the plane can be connected by at most three reflections. Equivalently, any plane isometry is a product of at most three reflections. This is the modern way of saying “the Euclidean group is generated by reflections,” and it is the engine behind every congruence theorem you ever learned.

Triangle congruence: classical criteria

For triangles, the abstract definition specialises to the famous side-and-angle tests:

SSS (Side-Side-Side): all three pairs of sides equal.

SAS (Side-Angle-Side): two sides and the included angle equal.

ASA (Angle-Side-Angle): two angles and the included side equal.

AAS: two angles and a non-included side equal (forced by SSS via the angle sum).

Each criterion is a recipe for constructing the isometry: locate one vertex, then a side, then the rest is forced.

Where this shows up

Manufacturing: Two machined parts are "interchangeable" if and only if they are congruent, same shape, same size, possibly reflected. ISO tolerance standards effectively define the allowable deviation from exact congruence.

Biochemistry: Chirality matters in biology: L-amino acids and D-amino acids are mirror-image congruent shapes but not isometry-congruent (one is not a translation/rotation of the other); life on Earth uses only L.

Computer Vision: Template matching tests SSS/SAS-style triangle congruence between landmark points in a stored model and detected features in an image; this is how early face-recognition algorithms worked.

The widget shows that any two congruent shapes can be connected by composing at most a few isometries. Try the “reflection then translation = glide reflection” preset, or build your own two-step composition that takes the F somewhere new.

Try it

Two triangles have sides $(3, 4, 5)$ and $(5, 3, 4)$ . Are they congruent? Which criterion?

Triangles with $\angle A = 60^\circ$ , $AB = 4$ , $\angle B = 80^\circ$ in both. ASA, congruent.

If two triangles have $AB = DE$ , $BC = EF$ , and $\angle A = \angle D$ , are they necessarily congruent? (Hint: the angle is not the included one.)

Construct an isometry that carries $\triangle ABC$ with $A = (0, 0)$ , $B = (3, 0)$ , $C = (0, 4)$ onto $\triangle A'B'C'$ with $A' = (1, 1)$ , $B' = (1, 4)$ , $C' = (-3, 1)$ . (Hint: it is a composition of a translation and a rotation.)

Pause: SSA (two sides and a non-included angle) is famously NOT a valid congruence criterion. Can you sketch two non-congruent triangles that share the same SSA data?

A trap to watch for

The SSA case is the chapter's longest-running trap. Many students assume that any three numerical facts force congruence, and the “ambiguous case” in trigonometry exists precisely because that intuition is wrong. With one fixed side, one fixed adjacent angle, and one fixed opposite side, you can sometimes swing the opposite side into two different positions, producing two genuinely different triangles. The fix: when you list three triangle facts, always note whether the angle is included between the two sides (SAS, valid) or not (SSA, sometimes valid, sometimes not). Lang spells the case out exactly because it is the classic algebra-versus-geometry gotcha, and you will see it again in the law of sines.

What you now know

You now have the formal definition of congruence as “there exists an isometry mapping one figure to the other,” and you know the relation behaves like equality. For triangles, the abstract criterion reduces to SSS / SAS / ASA / AAS, with SSA flagged as the exception. This closes Chapter 6: you can now move shapes around the plane with confidence and recognise when two configurations are genuinely the same up to rigid motion. The next chapter (“Distance and angle”) builds on these motions to develop the metric geometry of circles, lengths, and angles.

Quick check

Mark section complete →

References

Lang, S. (1971). Basic Mathematics. Springer. Chapter 6, §6, congruence defined via isometries, and SSS / SAS / ASA.

Euclid (c. 300 BCE). Elements, Book I, Propositions 4, 8, 26, the original SSS, SAS, and ASA theorems.

Hartshorne, R. (2000). Geometry: Euclid and Beyond. Springer. §9, modern axiomatic treatment of congruence and the SAS axiom.

Coxeter, H. S. M. (1969). Introduction to Geometry, 2nd ed. Wiley. Chapter 1: congruence by reflection and the “at most three reflections” theorem.

Greenberg, M. J. (2008). Euclidean and Non-Euclidean Geometries, 4th ed. W. H. Freeman. Chapters 3 and 9, the SSA ambiguous case and the limits of triangle congruence.

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