The Quadrilateral

Why does this exist?

A triangle is rigid. Fix three side lengths and the shape is locked — you proved that in the last lesson. Add one more point, one more side, and everything changes. The shape is no longer locked. It wobbles. It collapses. It can become almost anything.

Every special shape coming next — parallelogram, rectangle, rhombus, square — is just a quadrilateral with constraints added to stop the wobble. To understand those constraints, you first need to feel the wobble.

But here is the surprise. Even as the shape changes freely, one thing stays constant. Always. No matter what.

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Watch it move

One vertex is animated. The other three are fixed. Watch the four angles in the panel.

Canvas 400 320
Scale 50
Origin at 200 200

Point A at -3, -2
Point B at  3, -2
Point C at  2,  2
Point O at -1,  1
Circle K center O radius 1.8
Point D at -2,  2

Quadrilateral Q vertices A B C D
Color Q blue
Fill Q 0.08 blue

Angle alpha at A from D to B
Angle beta  at B from A to C
Angle gamma at C from B to D
Angle delta at D from C to A

Color alpha orange
Color beta  orange
Color gamma orange
Color delta orange

Show angle alpha
Show angle beta
Show angle gamma
Show angle delta

Animate D around Circle K speed slow

The four angles change constantly. But add them up — every single frame.

They sum to 360°. Every time.

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Try to break it

All four vertices are free now. Drag them anywhere. Make it convex. Make it squashed. Make it strange.

Canvas 400 340
Scale 50
Origin at 200 220
Grid 1

Point A at -3, -2
Point B at  3, -2
Point C at  2,  2
Point D at -2,  2

Quadrilateral Q vertices A B C D
Color Q blue
Fill Q 0.08 blue

Angle alpha at A from D to B
Angle beta  at B from A to C
Angle gamma at C from B to D
Angle delta at D from C to A

Color alpha orange
Color beta  orange
Color gamma orange
Color delta orange

Show angle alpha
Show angle beta
Show angle gamma
Show angle delta

Drag A
Drag B
Drag C
Drag D

You cannot break it. 360° — always.

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Why? One diagonal explains everything.

Draw a diagonal. One quadrilateral becomes two triangles. Each triangle carries 180°. Two triangles: 360°.

Canvas 400 300
Scale 50
Origin at 200 190

Point A at -3, -2
Point B at  3, -2
Point C at  2,  2
Point D at -2,  2

Triangle T1 vertices A B C
Triangle T2 vertices A C D
Fill T1 0.15 blue
Fill T2 0.15 orange
Color T1 blue
Color T2 orange

Segment AC from A to C
Style AC dashed

Label A "A"
Label B "B"
Label C "C"
Label D "D"

Show area T1
Show area T2

Drag A
Drag B
Drag C
Drag D

The diagonal AC splits the quadrilateral into triangle ABC and triangle ACD.

Triangle ABC: angles at A, B, C sum to 180°. Triangle ACD: angles at A, C, D sum to 180°.

Together: all four corners of the quadrilateral sum to 360°.

That is not a coincidence. That is a diagonal proof — one of the most useful tools in geometry. You will use this idea again and again in the lessons ahead.

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What you have discovered

A shape with four sides and four angles is called a quadrilateral. The word means exactly that — four sides.

The one property every quadrilateral shares, no matter its shape:

$\angle A + \angle B + \angle C + \angle D = 360°$

This is the angle sum property of a quadrilateral. It follows directly from the triangle angle sum — two triangles, two lots of 180°.

Everything that comes next in Branch 1 is a quadrilateral with something extra — parallel sides, equal sides, right angles. Each constraint removes some of the wobble. By the time you reach a square, all the wobble is gone and the shape is as rigid as a triangle.

But all of them are quadrilaterals first.