The angles of any quadrilateral

Try to draw a quadrilateral with three angles of 90° and a fourth that isn't 90°. However carefully you draw, the fourth corner either won't close, or it closes at exactly 90°.

Drawings can show you that something seems impossible, but only a reason tells you why. Here the reason is a property of every quadrilateral.

You already know the angles of a triangle add to 180°. A quadrilateral has a matching rule, and it comes straight from that one.

Take any quadrilateral (call it SOME) and draw the diagonal SM. That cuts it into two triangles, ∆SEM and ∆SOM.

In ∆SEM, ∠1 + ∠2 + ∠3 = 180°. In ∆SOM, ∠4 + ∠5 + ∠6 = 180°. Add all six: ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 180° + 180° = 360°.

Now regroup them as (∠1 + ∠4) + (∠3 + ∠6) + ∠2 + ∠5. The diagonal split two of the quadrilateral's corners into two parts, so each bracket is exactly one of those corners, and ∠2 and ∠5 are the other two corners, untouched. So these four terms are the four angles of the quadrilateral.

The angles of any quadrilateral add up to 360°. That settles the opening puzzle: if three angles are 90°, they use up 270°, leaving exactly 90° for the fourth. It has no choice.