Chapter 3.6 — Exercise 3.5 — SASAS Construction

Why a Chain of Three Sides and Two Angles Is Enough

Think of the construction as laying down three rigid rods end to end, with the angle between each consecutive pair fixed at a hinge. The first rod's position and direction are fixed the moment it's drawn. The angle at the first hinge fixes exactly where the second rod points, and its given length fixes exactly where it ends. The angle at the second hinge does the same for the third rod. By the time all three rods and both hinge angles are laid down, both open ends — the very start of the first rod and the very end of the third — sit at completely determined positions in space, which is why the segment joining them (the fourth side) is never in question and never needs a separate measurement — its length and direction are simply whatever the three rods and two hinges leave behind, in exactly the same way SSSDD's unlabelled fourth side is left to the diagonals to determine rather than being stated up front — two different methods arriving at the same underlying idea from different directions — one hands you diagonals to work from, the other hands you angles, but both leave exactly one side or vertex to fall out naturally, rather than requiring every single part of the shape to be stated in advance before a single line ever gets drawn on the page — the construction itself does the work of figuring out where that last side has to go, once the three known sides and two known angles have all been laid down correctly.

The General Method

Given three consecutive sides and the two angles between them, the quadrilateral builds up in a single connected chain, vertex by vertex:

• Draw the first side as a plain line segment.
• Draw a ray from its second endpoint at the first given angle.
• Mark the third vertex along that ray using the second side length as an arc radius.
• Draw a ray from that third vertex at the second given angle.
• Mark the fourth vertex along that ray using the third side length as an arc radius.
• Join the first and fourth vertices directly — this closing side is never measured, it simply falls out of where the chain of sides and angles ends up.

Worked Example

Construct quadrilateral PQRS with PQ = 3.6 cm, QR = 4.5 cm, RS = 5.6 cm, ∠RQP = 135° and ∠SRQ = 60°.

1. Draw line segment PQ of length 3.6 cm.
2. Draw a ray QX making a 135° angle with PQ.
3. With Q as centre, draw an arc of radius 4.5 cm, cutting QX at R.
4. Draw a ray RY making a 60° angle with QR.
5. With R as centre, draw an arc of radius 5.6 cm, cutting RY at S.
6. Join P–S to complete the required quadrilateral.

A Rhombus-Like Case and a Trapezium

Quadrilateral LAMP (AM = MP = PL = 5 cm, ∠M = 90°, ∠P = 60°): here three consecutive sides happen to share the same length, which doesn't change the method at all — the construction chain (side, angle, side, angle, side) runs exactly as before, just with the same radius reused for two of the three arcs.

Trapezium ABCD (AB ∥ CD, AB = 8 cm, BC = 6 cm, CD = 4 cm, ∠B = 60°): here only one angle is directly given, but the parallel-sides condition supplies the second. Since AB ∥ CD, the angles adjacent to side BC are supplementary: ∠C = 180° − ∠B = 180° − 60° = 120°. Once ∠C is known, the construction proceeds exactly like PQRS above — side, angle, side, angle, side, then join the ends.

Why the Closing Side Is Never Given

Every SASAS problem gives three sides and two angles — five measurements — and yet a quadrilateral has four sides in total. The fourth side, the one joining the very first and very last vertex in the chain, is deliberately left out, because once the other three sides and two angles are drawn correctly, that closing side's length and direction are already completely determined. Measuring it separately would be redundant information at best, and would risk contradicting the construction at worst if the stated length didn't quite match where the chain actually ends up. This is the same idea as SSSDD's unlabelled fourth side, just arrived at through angles and sides in sequence rather than through diagonals.

Recognising When a Property Supplies the Missing Angle

SASAS always needs two angles to complete the chain, but not every problem states both directly — the trapezium example shows how a shape's own definition (parallel sides forcing supplementary co-interior angles) can supply the second one. Before assuming an angle is genuinely unknown, check whether the shape's name already implies a relationship between it and another angle you do have.

Comparing SASAS to the Angle-Heavy Method Before It

SASAS and SASAA (from the previous exercise) both use three angles' worth of information in some form, but they're structured differently. SASAA scatters its three angles across three different vertices and needs the angle-sum property to find a fourth before drawing anything. SASAS instead concentrates its two angles at the two "joints" of a single connected chain of three sides, with the fourth side emerging automatically at the end — no angle-sum calculation is needed at all, because nothing here depends on knowing all four angles in advance. Recognising which pattern a problem matches — angles scattered around requiring a calculation, or angles sitting between consecutive sides in a chain — is what tells you which of the two methods to reach for. Both ultimately fix the same number of independent facts about the shape — five measurements' worth — just distributed differently across the four vertices.

Where the Chain of Sides and Angles Snaps

• Measuring the second angle from the wrong side. The second given angle is between the second and third sides specifically — measuring it against the first side instead produces a different (wrong) shape.
• Forgetting the closing side is unmeasured. The segment joining the first and last vertices is never given a length — it's simply drawn once both ends are correctly placed, not estimated or checked against a ruler beforehand.
• Missing a supplementary-angle relationship in trapezium-style problems, where only one of the two needed angles is stated directly and the other must be derived from the parallel-sides property first.

Everything Converging in One Final Method

The chain-of-sides-and-angles method from this exercise, along with the earlier diagonal-based methods, comes together in Exercise 3.6, which constructs rhombuses and squares directly from their diagonals using the perpendicular-bisector property. For the angle-sum reasoning this exercise builds on, revisit Exercise 3.4.