Quadrilaterals

42. The Parallelogram's Secrets · opposite sides equal, diagonals bisect

Opposite sides are equal, opposite angles are equal, adjacent angles sum to 180°, and diagonals bisect each other.

CDOAB = 304.1AB = 304.1BC = 270BC = 270CD = 304.1CD = 304.1DA = 270DA = 270OA = 219.3OA = 219.3OC = 219.3OC = 219.3OB = 186OB = 186OD = 186OD = 186∠DAB = 81°∠DAB = 81°∠BCD = 81°∠BCD = 81°
A parallelogram has four key properties: (1) opposite sides equal, (2) opposite angles equal, (3) adjacent angles supplementary (sum to 180°), and (4) diagonals bisect each other. Drag any vertex; all four properties hold simultaneously.

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Selina ICSE: Quadrilaterals

What this lesson covers

Try to break it

What happens if you drag A so close to B that the shape flattens? Does the parallelogram still hold its secrets?

How you build it

Construct a parallelogram.

  • Place point A — one corner of the parallelogram.
  • Place point C — the opposite corner. AC is one diagonal.
  • Join A and C — the first diagonal.
  • Place point B — a third corner, away from line AC.
  • Join A to B — one side of the parallelogram.
  • Join B to C — the next side.
  • With the Parallel tool, click point C, then click on side AB — it draws the line through C parallel to AB. Side CD lies along it.
  • With the Parallel tool, click point A, then click on side BC — it draws the line through A parallel to BC. Side DA lies along it.
  • Mark point D where the two parallel lines cross — your click snaps onto the exact point. D is the fourth corner.
  • Join B to D — the second diagonal. AC and BD cross at their shared midpoint, so the diagonals bisect each other.

The proof, step by step

Prove that a quadrilateral whose diagonals bisect each other is a parallelogram.

  • Consider triangles AOB and COD formed by the diagonals.
  • OA = OC and OB = OD because diagonals bisect each other by construction.
  • ∠AOB = ∠COD as they are vertically opposite angles.
  • By SAS congruence, △AOB ≅ △COD.
  • Therefore, AB = CD and ∠OAB = ∠OCD (alternate interior angles).
  • Similarly, △AOD ≅ △COB, so AD = BC.
  • Since alternate interior angles are equal, AB || CD and AD || BC.
  • Thus, opposite sides are equal and parallel, confirming all parallelogram properties.

Worked example

In parallelogram PQRS, diagonals intersect at O. If PO = 5 cm and QO = 3 cm, find the lengths of PR and QS.

Diagonals of a parallelogram bisect each other. So PR = 2 × PO = 2 × 5 = 10 cm. Similarly, QS = 2 × QO = 2 × 3 = 6 cm.

  • PR = 10 cm, QS = 6 cm — correct
  • PR = 5 cm, QS = 3 cm
  • PR = 15 cm, QS = 9 cm
  • PR = 10 cm, QS = 3 cm
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