247. Locking the Quadrilateral · 3 sides, 2 angles, one unique shape

Try to break it

Drag the side sliders (AB, BC, CD) and the angle dials (∠B, ∠C). A is fixed by AB and ∠B; D is fixed by CD and ∠C; the fourth side DA is simply whatever remains. Five measurements, one quadrilateral.

How you build it

Construct quadrilateral ABCD from its sides and angles.

• Point tool: mark the vertex B.
• Ray tool: click B, then click to the right — the base ray along which BC will lie.
• Arc tool: open the compass on the ruler — click the 0 mark, then the 5 mark — and the arc is drawn centred on B, cutting the base ray at C (BC = 5 cm).
• Point tool: mark C where the arc cuts the base ray.
• Angle tool: click B, click along BC toward C, then set ∠B = 120° with the −/+ buttons — the arm rises toward A.
• Arc tool: open the compass on the ruler — click the 0 mark, then the 4 mark — and the arc is drawn centred on B, cutting the ∠B arm at A (BA = 4 cm).
• Point tool: mark A where the arc cuts the ∠B arm.
• Angle tool: click C, click along CB toward B, then set ∠C = 100° with the −/+ buttons — the arm rises toward D.
• Arc tool: open the compass on the ruler — click the 0 mark, then the 4.5 mark — and the arc is drawn centred on C, cutting the ∠C arm at D (CD = 4.5 cm).
• Point tool: mark D where the arc cuts the ∠C arm.
• Segment tool: join D to A — the closing side. ABCD has AB = 4, BC = 5, CD = 4.5, ∠B = 120°, ∠C = 100°.

The proof, step by step

Prove that the constructed quadrilateral has the given three sides and two consecutive angles.

• We are given three sides (AB, BC, CD) and two consecutive angles (∠A, ∠B).
• Step 1: Draw BC. Step 2: Construct ∠B at B and draw a ray. Step 3: Mark A on the ray such that BA = given length.
• Step 4: Construct ∠A at A and draw a ray. Step 5: With C as centre and radius CD, draw an arc cutting the ray at D.
• Step 6: Join CD. The quadrilateral ABCD is now uniquely determined and satisfies all given conditions.

Worked example

In quadrilateral ABCD, AB = 4 cm, BC = 5 cm, CD = 4.5 cm, ∠B = 60° and ∠A = 120°. Which of the following sets of measurements would also uniquely determine a quadrilateral?

A quadrilateral is uniquely determined by 3 sides and 2 consecutive angles because the construction fixes two vertices, sets the included angle, fixes the third vertex, sets the next angle, and the final side length pins the fourth vertex at a unique intersection.

• 3 sides and 1 angle
• 3 sides and 2 consecutive angles — correct
• 4 sides and 1 angle
• 3 sides and 2 opposite angles