Congruence

Why does this exist?

Every proof in geometry that concludes "these two sides are equal" or "this angle equals that angle" is secretly running through one idea — congruence.

Measuring is not proof. You could measure two sides and find them equal, but how do you know the triangles are identical? Congruence tells you the minimum information that guarantees it. Once you have that, every matching part follows automatically — no further measuring needed.

That is why this exists. It is the engine inside most geometric proofs.

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The problem: a triangle is not pinned by one measurement

Two vertices are fixed. The third moves freely. Watch how many different triangles are possible.

Canvas 400 300
Scale 50
Origin at 200 200

Point A at -2, -1.5
Point B at 2, -1.5
Point O at 0, 0.5
Circle K center O radius 2.5
Point C at 0, 3

Triangle T vertices A B C
Color T blue
Fill T 0.08 blue

Show length AB inline
Show length BC inline
Show length CA inline

Animate C around Circle K speed slow

Infinitely many triangles — same base AB, completely different shapes.

So: what is the minimum information that pins a triangle down completely?

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Three sides are enough

Drag the left triangle. The right one is a live copy — built from the same three side lengths and nothing else.

Canvas 400 220
Scale 50
Origin at 200 200

Point A at -3.5, 0.5
Point B at -0.5, 0.5
Point C at -2, 2.5

Translate A by 4, 0 as P
Translate B by 4, 0 as Q
Translate C by 4, 0 as R

Triangle T1 vertices A B C
Triangle T2 vertices P Q R
Color T1 blue
Color T2 red
Fill T1 0.08 blue
Fill T2 0.08 red

Segment AB from A to B
Segment BC from B to C
Segment CA from C to A
Segment PQ from P to Q
Segment QR from Q to R
Segment RP from R to P

Mark equal AB PQ
Mark equal BC QR
Mark equal CA RP

Drag A
Drag B
Drag C

No matter how you drag — the copy follows perfectly. Three fixed side lengths leave no freedom whatsoever.

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Two sides and the angle between them

Now only two sides and the angle squeezed between them are fixed. Drag and watch.

Canvas 400 220
Scale 50
Origin at 200 200

Point A at -3.5, 0.5
Point B at -0.5, 0.5
Point C at -1, 2.5

Translate A by 4, 0 as P
Translate B by 4, 0 as Q
Translate C by 4, 0 as R

Triangle T1 vertices A B C
Triangle T2 vertices P Q R
Color T1 blue
Color T2 red
Fill T1 0.08 blue
Fill T2 0.08 red

Segment AB from A to B
Segment BC from B to C
Segment PQ from P to Q
Segment QR from Q to R

Mark equal AB PQ
Mark equal BC QR

Angle alpha at B from C to A
Angle beta at Q from R to P
Color alpha orange
Color beta orange
Mark equal-arcs alpha beta

Show angle alpha
Show angle beta

Drag A
Drag B
Drag C

The angle must sit between the two sides — not at a far corner. That position is everything. Move it to a far corner and the test fails.

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Two angles and the side between them

Two angles fix the shape of the triangle. The side between their vertices fixes the size. Together: one unique triangle.

Canvas 400 220
Scale 50
Origin at 200 200

Point A at -3.5, 0.5
Point B at -0.5, 0.5
Point C at -1, 2.5

Translate A by 4, 0 as P
Translate B by 4, 0 as Q
Translate C by 4, 0 as R

Triangle T1 vertices A B C
Triangle T2 vertices P Q R
Color T1 blue
Color T2 red
Fill T1 0.08 blue
Fill T2 0.08 red

Segment AB from A to B
Segment PQ from P to Q
Mark equal AB PQ

Angle alpha at A from B to C
Angle beta at B from C to A
Angle gamma at P from Q to R
Angle delta at Q from R to P
Color alpha orange
Color beta orange
Color gamma orange
Color delta orange
Mark equal-arcs alpha gamma
Mark equal-arcs beta delta

Show angle alpha
Show angle beta

Drag A
Drag B
Drag C

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Right angle, hypotenuse, one side

Right triangles earn a special test. The right angle is already given — that counts as a free angle. So hypotenuse plus one leg is enough.

Canvas 400 220
Scale 50
Origin at 200 200

Point A at -3.5, 0.5
Point B at -0.5, 0.5
Point C at -0.5, 2.5

Translate A by 4, 0 as P
Translate B by 4, 0 as Q
Translate C by 4, 0 as R

Triangle T1 vertices A B C
Triangle T2 vertices P Q R
Color T1 blue
Color T2 red
Fill T1 0.08 blue
Fill T2 0.08 red

Mark right-angle at B
Mark right-angle at Q

Segment AB from A to B
Segment AC from A to C
Segment PQ from P to Q
Segment PR from P to R

Mark equal AB PQ
Mark equal AC PR

Drag A
Drag B
Drag C

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What you have discovered

Two triangles with exactly the same shape and size are called congruent. We write:

$\triangle ABC \cong \triangle PQR$

The order of vertices matters — A matches P, B matches Q, C matches R.

The four minimum tests that guarantee congruence:

Test	What pins the triangle down
SSS	Three pairs of equal sides
SAS	Two sides + the included angle
ASA	Two angles + the included side
RHS	Right angle + hypotenuse + one leg

Once you establish congruence, every matching part is automatically equal — sides, angles, everything. This is called CPCT: Corresponding Parts of Congruent Triangles.

Prove two triangles congruent once. Then claim any matching pair for free. That is how geometry avoids measuring everything.