Chapter 3.2 — Exercise 3.1 — SSSSA Construction

The General Method

For a quadrilateral ABCD with AB, BC, CD, DA given as the four sides and ∠A given as the one angle:

• Draw the base side — the side adjacent to the given angle — as a plain line segment.
• Draw a ray from the angle's vertex at the given angle to the base, which is where the next vertex will eventually be marked.
• Swing an arc using the next known side as radius, centred at the vertex the ray starts from, to mark that next vertex on the ray.
• Swing two more arcs — one from each of the two vertices already placed, using the two remaining side lengths as radii — and mark where they cross as the final vertex.
• Join the final vertex to both of the points whose arcs located it, completing the quadrilateral.

Worked Example — A General Quadrilateral

Construct quadrilateral ABCD with AB = 5.5 cm, BC = 3.5 cm, CD = 4 cm, AD = 5 cm and ∠A = 45°.

1. Draw line segment AB of length 5.5 cm.
2. Draw a ray AX making a 45° angle with AB.
3. With A as centre, draw an arc of radius 5 cm, cutting AX at D.
4. With D as centre, draw an arc of radius 4 cm.
5. With B as centre, draw an arc of radius 3.5 cm, cutting the previous arc at C.
6. Join B–C and D–C to complete quadrilateral ABCD.

The Same Method Applied to Special Quadrilaterals

The rest of this exercise applies this exact method to a parallelogram, a rhombus, a rectangle, and a square — the only thing that changes each time is how many of the four side lengths you actually need to be told, because the shape's own properties supply the rest.

• Parallelogram PQRS (PQ = 4.5 cm, QR = 3 cm, ∠RQP = 60°): since opposite sides of a parallelogram are equal, PQ = 4.5 cm tells you RS = 4.5 cm too, and QR = 3 cm tells you PS = 3 cm — only two side lengths were actually given, but all four are known before drawing begins.
• Rhombus MATH (AT = 4 cm, ∠MAT = 120°): since all four sides of a rhombus are equal, one given side length (AT = 4 cm) fixes every side — MA = TH = MH = 4 cm as well.
• Rectangle FLAT (FL = 5 cm, LA = 3 cm): opposite sides equal gives AT = 5 cm and FT = 3 cm, and every angle in a rectangle is automatically 90°, so the "one angle" of SSSSA is already known without being stated.
• Square LUDO (LU = 4.5 cm): all four sides equal means one measurement fixes the entire square, and again every angle is automatically 90°.

In every one of these four cases, the construction steps themselves are identical to the general ABCD example — draw the base, mark the angle, swing three arcs. The only work happening beforehand is reasoning out the side lengths and angle that the shape's name already guarantees, before a single arc is drawn.

Why the Construction Produces Exactly One Shape

It's worth pausing on why this sequence of steps can't accidentally produce two different quadrilaterals with the same five measurements. The base side and the angle at A fix the direction of AD precisely, and the arc from A fixes D's exact position along that direction. From there, C has to sit somewhere that is simultaneously 3.5 cm from B and 4 cm from D — two circles centred at B and D, and two circles generally cross at exactly one point on each side of line BD. Since a quadrilateral's vertices are conventionally read in one consistent order (say, anticlockwise), only one of those two crossing points gives a valid, non-self-crossing quadrilateral — which is exactly the point the construction steps select.

Reading the Given Information Before Starting

The real skill this exercise builds isn't compass technique — it's recognising, before drawing anything, exactly which of the four sides and the one angle you've actually been handed, and which ones a property (equal opposite sides, all sides equal, all angles 90°) fills in for you. Skipping this step and trying to draw directly from an incomplete set of measurements is the most common way this construction goes wrong.

Checking a Finished Construction

Once ABCD is drawn, it's worth measuring the sides that weren't directly used as the last arc's radius, as a check. BC and CD were both used to place C, so they're guaranteed correct by construction — but re-measuring AD against the intended 5 cm, and confirming ∠A really reads 45° on a protractor, catches any drift that crept in from an imprecisely set compass width or a ray drawn a degree or two off. This kind of check matters more in constructions than in most other areas of mathematics, since a small physical error compounds silently until the final shape is visibly off, without any single step looking obviously wrong. Building this habit of checking the "unused" measurement, rather than only the ones that guided the drawing, catches errors early instead of at the very end. It's the geometric equivalent of substituting a solved equation's answer back into the original — confirming the whole system is consistent, not just the last step performed — a small habit that catches drift before it becomes a visibly wrong shape, and one worth applying to every construction in this chapter, not only this first one, since the same two-arcs-locate-a-vertex idea repeats in every method that follows, just with a diagonal or an angle occasionally standing in for one of the arcs, depending on which five measurements a given problem happens to hand you rather than the ones used here.

Slips That Show Up at the Compass, Not the Concept

• Drawing the angle at the wrong vertex. The given angle belongs to a specific vertex of the base side — check which one before drawing the ray.
• Confusing which sides are still unknown after using a shape's properties to fill in the rest — write out all four side lengths explicitly before starting to construct.
• Losing arc precision. Since the final vertex is located by two arcs crossing, a compass that slips even slightly changes where they intersect — keep the compass width fixed once set for each radius.

The Same Idea, With a Diagonal Swapped In

This same three-arcs-from-two-known-points idea reappears with a diagonal in place of one of the sides in Exercise 3.2, and with two diagonals in Exercise 3.3. For the underlying properties this exercise leans on — which quadrilateral guarantees which equal sides or right angles — revisit the Introduction to Quadrilaterals.