Foundations — Quadrilaterals — square, rectangle, parallelogram, rhombus, trapezium, kite

Before you can read a single line of the parent note, you need to own every symbol it throws at you. This page builds them from nothing, in an order where each one leans only on the ones before it. If the parent note said "draw diagonal $AC$" or "$\angle A+\angle B=180°$" and you froze — this is the page that unfreezes you.

1. A point, and how we name it

Look at the four amber dots in the figure below. Each is a "post" of our fence. The letter is only a name tag — it lets us talk about "the corner called $B$" instead of pointing.

Why the topic needs it: every quadrilateral is described by its four corner points. When the parent writes $△ABC$ or "side $AB$", it is stringing point-names together — so the names must mean something first.

2. A line segment, and the symbol $AB$

So $AB$ does double duty:

• The stick itself (the drawn edge from $A$ to $B$).
• Its length (a plain number, like $5\text{cm}$).

Context tells you which. In "$AB=CD$" both are lengths, so the sentence reads "the stick from $A$ to $B$ is the same length as the stick from $C$ to $D$."

Why the topic needs it: a quadrilateral's four sides are four segments: $AB$, $BC$, $CD$, $DA$. "Opposite sides equal" is the sentence $AB=CD$.

See Coordinate Geometry - Distance Formula for turning two corner positions into an actual length number.

3. Going around the loop: naming a quadrilateral

The arrows show the walk. Notice the sides come from consecutive letters, and the two "opposite" sides are the ones you meet on opposite legs of the walk: $AB$ opposite $CD$, and $BC$ opposite $DA$.

4. The angle symbol $\angle$ and "$90°$"

Reference the figure:

• The cyan angle at $B$ is a wide, lazy opening.
• The amber angle marked with a tiny square is exactly $90°$ — a right angle. That little square is the universal "this is a perfect corner" symbol.

Some milestones you'll meet:

• $90°$ = right angle = the square corner of a wall or a screen.
• $180°$ = a straight line (a "corner" that's been flattened out completely).
• $360°$ = one full turn.

Why the topic needs it: "rectangle = all four angles $90°$" and "angle sum $=360°$" are the two headline facts of the whole chapter. Both are sentences about $\angle$ and $°$.

5. Parallel sides and the symbol $\parallel$

The two cyan tracks stay a constant gap apart. The amber pair tilt toward each other — extend them and they'd cross, so they are not parallel.

Why the topic needs it: parallelism is the sorting rule for the whole quadrilateral family:

• both pairs of opposite sides parallel → parallelogram,
• exactly one pair parallel → trapezium,
• no pairs parallel → a general quadrilateral or kite.

Deep dive on how parallel lines create equal angles: Basic Geometry - Parallel Lines.

6. Equal marks — tick marks on sides and arcs on angles

This is a visual shorthand so a diagram can say "$AB=CD$" and "$\angle A=\angle C$" without any words. When the parent note shows a rhombus with a single tick on all four sides, it's screaming "all sides equal" in one glance.

7. The diagonal

Trace it: $A$ and $C$ are opposite corners (not neighbours), so $AC$ leaps across the interior. Same for $BD$. The point where the two diagonals cross is usually named $O$.

Why the topic needs it: every derivation in the parent note starts with "draw a diagonal". It converts an unfamiliar four-sided shape into familiar triangles — see Basic Geometry - Triangles.

8. Base, height, and the perpendicular symbol

Study the figure hard, because this is the single most-failed idea in the topic:

• The amber dashed line is the height $h$ — it stands straight up ($90°$, note the square) from the base.
• The cyan slanted side is longer than $h$. It is not the height.

Why the topic needs it: area of parallelogram $=b\times h$ and trapezium $=\frac{1}{2}(a+b)h$ both use the perpendicular $h$. Confuse it with the slant and every area is wrong.

9. The letters in the area formulas

Now that pictures back every idea, the formula symbols are just labels for measured numbers:

The little raised $2$ in ${\text{cm}}^2$ means "square centimetres" — the count of $1\text{ cm}\times 1\text{ cm}$ tiles that fit inside. More on area vs. perimeter in Mensuration - Area and Perimeter.

10. Two Greek/math tools the proofs quietly assume

The parent proves "opposite sides equal" using the ASA rule (Angle–Side–Angle: if two triangles share one equal side sitting between two equal angles, they're congruent). That machinery lives in Basic Geometry - Triangles; and the side-length calc in Example 3 uses Basic Geometry - Pythagorean Theorem.

How these foundations feed the topic

Read it top-down: raw points become segments, four segments make the loop, and angles plus parallelism describe it. The diagonal is the hub — it powers both the angle-sum fact and the congruence proofs, while base/height and diagonals power the area side.

Equipment checklist

Cover the right-hand side and see if you can state each from memory.