Construction of Polygons [Using Ruler and Compass Only]

248. The Parallelogram Blueprint · Two sides and an angle are all you need

Opposite sides remain equal and parallel as you drag.

BDAB = 150AB = 150BC = 200BC = 200CD = 150CD = 150DA = 200DA = 200∠B = 53°∠B = 53°AC
A parallelogram is determined by two adjacent sides and the angle between them, because opposite sides are equal and parallel. Once one corner is set, the opposite sides simply copy the first pair.

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Selina ICSE: Construction of Polygons [Using Ruler and Compass Only]

What this lesson covers

Try to break it

Drag A along the base line to set AB's length, and C around B's circle to set the angle at B. The opposite sides AD and BC always come out equal — the construction guarantees a parallelogram for every choice of base length and base angle.

How you build it

Construct parallelogram ABCD with AB = 5 cm, BC = 7 cm and ∠B = 60°.

  • Point tool: mark the vertex B — the corner that holds the two sides and the angle.
  • Ray tool: click B, then click out to the left — the base ray along which side BA will lie.
  • Arc tool: open the compass on the ruler — click the 0 mark, then the 5 mark — centred on B, the arc cuts the base ray at A (BA = 5 cm).
  • Point tool: mark A where the arc cuts the base ray.
  • Angle tool: click B, click along the base ray, then set ∠B = 60° with the −/+ buttons — the second arm rises toward C.
  • Arc tool: open the compass on the ruler — click the 0 mark, then the 7 mark — centred on B, the arc cuts the ∠B arm at C (BC = 7 cm).
  • Point tool: mark C where the arc cuts the ∠B arm.
  • Arc tool: open the compass to 7 cm on the ruler, centre A — AD must equal BC, so swing an arc of 7 cm.
  • Arc tool: open the compass to 5 cm on the ruler, centre C — CD must equal AB; this arc cuts the previous one at D.
  • Point tool: mark D where the two arcs cross — opposite sides now copy the first pair.
  • Segment tool: join A to D.
  • Segment tool: join C to D — parallelogram ABCD is complete (AB = CD = 5, BC = AD = 7).

The proof, step by step

Prove that the constructed figure is a parallelogram with the given sides and angle.

  • AB = CD (by construction, both equal to 'a')
  • BC = AD (by construction, both equal to 'b')
  • ABCD is a parallelogram (opposite sides are equal)

Worked example

In parallelogram ABCD, AB = 5 cm, BC = 7 cm, and angle B = 45°. What is the length of CD?

In a parallelogram, opposite sides are equal in length. Since AB = 5 cm, the opposite side CD must also be 5 cm.

  • 3 cm
  • 5 cm — correct
  • 7 cm
  • 12 cm
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