Quadrilaterals

Floor tiles, windows, books, screens — look around and the most common man-made shape is not the triangle but the quadrilateral. One extra side over a triangle, and a whole extra dose of character.

Four sides, four angles, 360°

A quadrilateral has 4 sides, 4 corners and 4 interior angles. Its angle sum needs no memorizing: draw one diagonal and the shape splits into two triangles, each summing to 180° (fresh from Triangles):

$2 \times 180^\circ = 360^\circ$

So the four angles of any quadrilateral add up to exactly 360° — one full turn, no more, no less.

The quadrilateral family tree

Quadrilaterals form a family where special members nest inside general ones, each generation adding a new rule:

Trapezoid: at least one pair of parallel sides — the family's most free-spirited member;
Parallelogram: both pairs of opposite sides parallel and equal;
Rectangle: a parallelogram with four right angles;
Rhombus: a parallelogram with four equal sides;
Square: four equal sides and four right angles — a special rectangle and a special rhombus at once;
Kite: two pairs of neighboring sides equal, with perpendicular diagonals.

The square is the family's model student: it inherits everything a rectangle has, everything a rhombus has, and keeps it all.

A small dispute of definitions

A trapezoid has two common definitions: "exactly one pair of parallel sides" or "at least one pair". Under the first, a parallelogram is not a trapezoid; under the second, it counts as a special one. Both textbooks exist — check which one yours uses before homework.

InteractiveAngle Explorer

Angle60

Type：Acute= 1.047 rad

180° = π rad

Drag the slider and watch the angle change: under $90^\circ$ it is acute, exactly $90^\circ$ right, and past that obtuse. A quadrilateral may hold no right angle at all, or four of them — but the four angles must always total 360°.

Hunting the missing angle

Know three angles and the fourth falls out at once. A quadrilateral has angles of $90^\circ$ , $80^\circ$ and $110^\circ$ :

$360^\circ - (90^\circ + 80^\circ + 110^\circ) = 360^\circ - 280^\circ = 80^\circ$

The fourth angle is $80^\circ$ . Check: $90 + 80 + 110 + 80 = 360$ — a full turn, as promised.

Pair first, subtract last

Do not grind through the sum head-on: $90 + 110 = 200$ is instant, $200 + 80 = 280$ rolls off the tongue, and then $360 - 280 = 80$ . Making friendly sums first keeps the whole thing mental.

A family that loves to flex

Fix a triangle's sides and its shape is locked; quadrilaterals, though, perform tricks — a parallelogram frame skews at the lightest push. That is a flaw and a talent: scissor gates and folding clothes racks earn their living on exactly this flexibility. And when a frame must hold still, add one diagonal brace and split it into two triangles. Flexibility and rigidity — the two families complete each other.

Check yourself

Quick quiz

1. What is the angle sum of a quadrilateral?
2. Three angles are 90°, 80° and 110°. What is the fourth?
3. Which statement is true?