The folded triangle,
angle by angle
PSLE 2017 Paper 2, Question 17 is a favourite “hardest question” for a reason: it chains four separate ideas — isosceles base angles, the angle sum of a triangle, angles on a straight line, and what folding actually does to an angle. Get comfortable with the chain here, and every fold-and-angle question after this gets easier.
📝 PSLE 2017, Paper 2, Question 17
Minah has a triangular piece of paper ABC with BA = BC, ∠ABC = 84° and ∠CDE = 67°. ADC and BEC are straight lines. She folds the paper along the line DE, as shown below.
(a) Find ∠x. (b) Find ∠y.
💡 How to start any folding question
Folding is a reflection in the crease DE. Paper doesn’t stretch, so every angle inside the flap is copied exactly — it just lands somewhere new.
That means the first move is never a calculation. It is one of these two questions:
- Is the unknown a folded copy of something? If so, go back to the before picture and find the original instead. (This solves ∠x.)
- If not, which triangle does it sit in, and what is the one missing angle there? Then go and get that angle. (This solves ∠y.)
Part (a): Find ∠x
2 marks. Before touching any numbers, work out what ∠x actually is.
🔍 Start at the end: what is ∠x?
∠x sits at E, between the crease ED and the folded edge EC′. Before the fold, that same angle was ∠DEC — the angle at E inside triangle CDE. Folding just carried it across the crease unchanged. So the question “find ∠x” is really the question “find ∠DEC”, and that is an ordinary triangle problem. Everything below is just getting there.
-
1
Rewrite the question: ∠x = ∠DEC.
Folding is a reflection in DE, and reflections don’t change the size of an angle. So instead of chasing ∠x in the messy folded picture, go back to the clean before picture and find ∠DEC there.
-
2
To use triangle CDE, first find ∠DCE.
Triangle CDE already gives us ∠CDE = 67°. We need one more angle. Since BA = BC, triangle ABC is isosceles, so its base angles are equal: ∠BAC = ∠BCA = (180° − 84°) ÷ 2 = 48°. D lies on AC, so ∠DCE is that same angle: 48°.
-
3
Angle sum of triangle CDE.
∠DEC = 180° − 48° − 67° = 65°.
-
4
Carry it back across the fold.
∠x is that angle reflected, so ∠x = ∠DEC = 65°.
Answer — part (a)
∠x = 65°
Part (b): Find ∠y
3 marks — and again, the first move is working out which angle would unlock it.
🔍 Start at the end: what would unlock ∠y?
Look at where ∠y sits. The folded edge DC′ crosses edge AB — call that crossing point F (the exam paper doesn’t name it, so mark it in yourself). ∠y is ∠AFD, an angle inside triangle ADF.
In that triangle we already know ∠DAF = ∠BAC = 48°. So there is exactly one missing piece: ∠ADF — which is ∠C′DA, the angle the folded edge makes with DA. Find ∠C′DA and ∠y drops out of the angle sum. That is the whole plan.
-
1
Identify the triangle ∠y lives in.
Mark F where DC′ crosses AB. Then ∠y = ∠AFD, inside triangle ADF, whose other two angles are ∠DAF and ∠ADF (= ∠C′DA).
-
2
One angle of that triangle is already known.
∠DAF is just the original base angle at A, untouched by the fold: 48°.
-
3
Unlock the missing angle, ∠C′DA.
ADC is a straight line, so the angles at D add to 180°. Turning from DC round to DA you pass through two equal 67° angles: ∠CDE = 67° (given), and ∠EDC′ = 67° (its mirror image, because the fold copied it across the crease). What is left over is ∠C′DA:
∠C′DA = 180° − 67° − 67° = 46°
-
4
Angle sum of triangle ADF.
∠y = 180° − 48° − 46° = 86°.
Answer — part (b)
∠y = 86°
Four skills, chained in one question
This is what pushes a question to 5 marks and a top difficulty rating — not any single step, but needing all four without a prompt.
- ✓Isosceles triangle base angles.BA = BC tells you two angles are equal before you're told anything about D or E.
- ✓Angle sum of a triangle (twice).Once in triangle CDE for part (a), and again in the unlabelled triangle ADF for part (b).
- ✓Angles on a straight line.ADC being straight is what lets you strip off the two 67° angles at D to reach 46° in part (b).
- ✓Recognising what a fold preserves.Angles and lengths carry over unchanged — only their position on the page changes.
★ Teacher’s power tip
Notice that neither part of this question starts with a calculation. Both start by naming the unknown: ∠x is ∠DEC folded over; ∠y is the third angle of triangle ADF. Once the unknown has a name, the arithmetic is P5 work. Students who dive straight into computing angles may produce three correct numbers that lead nowhere.
A useful habit: whenever a folded edge crosses another edge without a labelled point there, mark that crossing yourself and give it a letter (like F here). You cannot name the triangle your unknown sits in until its corners have names.