Chapter 3.5 — Exercise 3.4 — SASAA Construction

From Hinges to Angles: Why This Method Needs Three

Two adjacent sides alone, like a hinge joint, can open to any angle at all — nothing about the two lengths HE and EL by themselves fixes how wide angle E is. Giving ∠E removes that one degree of freedom, fixing the shape of the first triangle-like piece HEL exactly. But that still leaves the rest of the quadrilateral free to swing around L and H independently, which is precisely what the two remaining angles, ∠H and ∠P, pin down. Three angles, alongside the two sides that start the shape off, is the minimum needed to remove every remaining flex — which is also exactly why the fourth angle, ∠L, doesn't need to be given separately: by the time three angles and two sides have removed all the flex, ∠L is already forced to be whatever value keeps the total at 360° — there's no remaining freedom left for it to be anything else — the calculation is really just naming a value that the rest of the construction has already settled, before a single ray has even been drawn, which is exactly why skipping it and guessing at ∠L instead never ends well, no matter how close the guess happens to look on paper next to the other three angles measured out around the shape — a protractor reading that's even a degree or two off compounds into a visibly wrong final vertex once both closing rays are drawn.

Finding the Missing Angle First

Whenever three angles of a quadrilateral are given, the fourth follows directly from the angle-sum property:

This calculation always comes before any drawing — trying to construct the shape without first knowing all four angles leaves the last vertex impossible to close off correctly.

The General Method

For a quadrilateral HELP with sides HE, EL given and angles ∠H, ∠E, ∠P given (so ∠L is calculated first):

• Draw the shared side between the two given sides as a plain line segment.
• Draw a ray from one end at the angle given for that vertex.
• Mark the second vertex along that ray using the second given side length as an arc radius.
• Draw two more rays — one from the far end of the base at its given angle, and one from the newly marked vertex at the angle adjacent to the second side — and let them intersect to locate the final vertex.

Worked Example

Construct quadrilateral HELP with HE = 6 cm, EL = 4.5 cm, ∠H = 60°, ∠E = 105° and ∠P = 120°.

First, the missing angle: ∠L = 360° − (60° + 105° + 120°) = 360° − 285° = 75°.

1. Draw line segment HE of length 6 cm.
2. Draw a ray EX making a 105° angle with HE.
3. With E as centre, draw an arc of radius 4.5 cm, cutting EX at L.
4. Draw a ray HY making a 60° angle with HE.
5. Draw a ray LZ making a 75° angle with EL, cutting ray HY at P.
6. HELP is the required quadrilateral.

Applying It to a Parallelogram and a Rectangle

Parallelogram GRAM (GR = AM = 5 cm, RA = MG = 6.2 cm, ∠R = 85°): opposite sides of a parallelogram are equal, so only two side lengths are ever really independent here. For the angles, opposite angles are equal (∠R = ∠M = 85°) and adjacent angles are supplementary, so ∠G = ∠A = 180° − 85° = 95°. All three angles this method needs are known immediately from a single given angle, without any additional information.

Rectangle FLAG (FL = 6 cm, LA = 4.2 cm): opposite sides equal gives AG = 6 cm and FG = 4.2 cm, and every angle in a rectangle is 90° by definition — so all three angles the SASAA method calls for are already known before construction starts, with nothing left to calculate.

Why Three Angles Are Needed, Not Two

It might seem like giving the fourth angle up front, rather than making you calculate it, would save a step — but the calculation itself is doing real work. Three angles plus the two adjacent sides is exactly the amount of information needed to fix the quadrilateral's shape uniquely; two angles alone would leave the far side of the shape free to swing into more than one possible position, since nothing would constrain where the last ray has to point. Computing the fourth angle from the angle-sum property isn't a formality before construction — it's the piece of information that makes the shape rigid rather than flexible.

Why the Order of Rays Matters

Each ray in this construction is drawn from a specific vertex at a specific angle to a specific existing side — swapping which side an angle is measured against changes where the resulting ray points, even if the angle's numeric value is correct. Before drawing any ray, it helps to state explicitly "this angle is measured from this side, at this vertex" — a habit that prevents the two final rays from being drawn in directions that never actually meet.

Where the Two Given Sides Sit

Notice that the two given sides in this method — HE and EL for quadrilateral HELP — are adjacent, sharing the vertex E between them, not opposite sides across the shape. This isn't incidental: the construction chain starts at H, runs along the shared side HE to E, and then needs EL to reach the third vertex L before the two closing rays can be drawn from H and L. Two opposite (non-adjacent) sides wouldn't connect into a single chain this way, which is exactly why every SASAA problem in this exercise specifies two sides meeting at a common vertex. Notice, too, which vertex's angle is always the one calculated rather than given: it's L, the vertex diagonally opposite the shared vertex E — the farthest point from where the two known sides meet, and the last one the construction chain reaches.

Angle Slips Worth Watching For

• Skipping the fourth-angle calculation. Always compute the missing angle from the angle-sum property before drawing a single line — attempting the construction with only three known angles and a guess at the fourth doesn't work.
• Measuring an angle from the wrong side. Each angle in this method is defined relative to a specific side meeting at that vertex — double-check which side before placing the protractor.
• Forgetting that a parallelogram's angles are already linked. One given angle fixes all four angles at once through the equal-opposite and supplementary-adjacent rules — there's no need for three separate angle measurements in these special cases.

Trading Two Sides for Three

The angle-sum calculation from this exercise carries directly into Exercise 3.5, where three sides and two included angles replace two sides and three angles. For the diagonal-based methods that precede this one, revisit Exercise 3.3.