Chapter 3.1 — Introduction to Quadrilaterals

Convex and Concave — Which One This Chapter Covers

A quadrilateral is convex if a line joining any two of its interior points stays entirely inside the figure — every interior angle is less than 180°, and both diagonals lie within the shape. A quadrilateral is concave if at least one interior angle is a reflex angle (greater than 180°), which lets a line joining two interior points pass outside the figure — one diagonal ends up outside the shape entirely. This chapter's construction methods apply only to convex quadrilaterals; every square, rectangle, parallelogram, rhombus, and trapezium you construct in the exercises ahead falls into this category.

How the Special Quadrilaterals Are Related

Quadrilaterals are further classified by how many sides are parallel and whether their sides or angles are equal. Each type in this list adds one more condition on top of the last:

• Trapezium — exactly one pair of opposite sides is parallel (AB ∥ DC). The angles adjacent to each non-parallel side are supplementary: ∠A + ∠D = 180° and ∠B + ∠C = 180°.
• Parallelogram — both pairs of opposite sides are parallel. This adds: opposite sides equal (AB = DC, AD = BC), opposite angles equal (∠A = ∠C, ∠B = ∠D), adjacent angles supplementary, and diagonals that bisect each other (OA = OC, OB = OD).
• Rectangle — a parallelogram with one right angle. Because a parallelogram's angles come in supplementary pairs, one right angle forces all four to be 90°. A rectangle also gains a property no general parallelogram has: its diagonals are equal in length (AC = BD), not just bisecting.
• Rhombus — a parallelogram with all four sides equal. Its distinguishing extra property is that its diagonals bisect each other at right angles (∠AOB = 90°), not merely bisecting like an ordinary parallelogram's.
• Square — satisfies both the rectangle condition (all angles 90°) and the rhombus condition (all sides equal) at once. Its diagonals combine both special properties: equal in length, and bisecting each other at right angles.

Seen this way, a square isn't really a separate shape from the others — it's a parallelogram with every extra condition applied simultaneously. Recognising which of these properties a problem hands you is exactly how you'll decide which construction method to use in each exercise — a rhombus's equal sides and perpendicular diagonals point toward a very different starting move than a trapezium's single pair of parallel sides.

Why Five Measurements Are Always Enough

A quadrilateral has four sides, four angles, and two diagonals — ten quantities in total — but constructing one uniquely never requires all of them. Every exercise in this chapter gives exactly five independent measurements, drawn from different combinations: four sides and one angle, four sides and one diagonal, three sides and two diagonals, two sides and three angles, or three sides and two angles. Five is the minimum that pins down a convex quadrilateral's exact shape and size — fewer than five leaves the shape ambiguous, since a hinge-like quadrilateral can flex into more than one form even when some measurements match.

Naming a Quadrilateral in the Right Order

Writing "quadrilateral ABCD" isn't just a label — the letter order tells you exactly which vertices are adjacent and which are opposite. Going around ABCD in order, the sides are AB, BC, CD and DA; the diagonals connect the two pairs of letters that are not next to each other in that sequence, namely AC and BD. Get the order wrong — say, writing ABDC for the same physical shape — and AB stops being a side and instead becomes a diagonal, since B and D are no longer adjacent in that naming. This matters constantly in the exercises ahead, where a construction depends entirely on correctly identifying which given measurement is a side and which is a diagonal.

Why the Angle-Sum Property Is Always True

The fact that any quadrilateral's four angles sum to 360° isn't a rule to memorise in isolation — it follows directly from something already familiar: a triangle's angles always sum to 180°. Draw either diagonal of a quadrilateral, and it splits the shape into exactly two triangles. Each triangle contributes 180° of angle, and together their angles account for all four of the quadrilateral's angles exactly once, giving 180° + 180° = 360° in total. Seeing why a property is true, rather than only what it states, is what makes it usable in situations the rule itself never explicitly mentions.

Common Misconceptions to Avoid

• A rhombus is not automatically a square. All four sides being equal only guarantees a rhombus — the angles still need to be checked separately for 90°.
• Equal diagonals are not a general parallelogram property. Only rectangles and squares have equal diagonals; an ordinary parallelogram's diagonals bisect each other but are usually different lengths.
• Skipping the angle-sum check. Before attempting any construction where three angles are given, always compute the fourth using ∠A + ∠B + ∠C + ∠D = 360° first — attempting to construct without it is a common source of errors.

Sides, Angles, and Diagonals — Ten Quantities, Not All Independent

It's tempting to think of a quadrilateral's four sides, four angles, and two diagonals as ten separate, unrelated numbers, but they're tightly linked. The four angles are never independent of each other — fix any three and the fourth is forced by the 360° total. The diagonals aren't independent of the sides either: once all four sides and one diagonal are fixed, the second diagonal's length is already determined, even before it's drawn, because the triangle it would complete is already fully specified. This interconnectedness is precisely why five measurements, chosen well, are always enough to fix the remaining five — nothing genuinely new is left to specify once the right five are known.

From Properties to Constructions

Every property covered here — the angle sum, the parallel-side conditions, and especially how a parallelogram's, rectangle's, rhombus's, and square's diagonals each behave differently — becomes the reasoning behind a specific construction technique in the exercises ahead, starting with Exercise 3.1, where four sides and one angle are used to build a quadrilateral from scratch. These same properties reappear in more depth later on, with formal proofs replacing construction steps, in Quadrilaterals in Class 9.