Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium]

236. The Parallelogram Key · One pair equal & parallel unlocks the shape

Opposite sides AD and BC remain equal and parallel as you drag.

C∠3∠4AD = 223.6AD = 223.6BC = 223.6BC = 223.6∠1 = 34°∠1 = 34°∠2 = 34°∠2 = 34°ABD
A quadrilateral is a parallelogram if one pair of opposite sides is both equal and parallel. Once AD = BC and AD ∥ BC, the figure is forced to be a parallelogram, so the other pair of sides is equal and parallel as well.

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Selina ICSE: Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium]

What this lesson covers

Try to break it

Drag A, B, or D. The construction holds AB = DC and AB ∥ DC at all times — and the moment those two conditions are true on one pair of sides, the other pair (AD and BC) automatically becomes parallel and equal too. Try to find a shape where AD and BC tilt apart; impossible.

How you build it

Construct a parallelogram ABCD.

  • Point tool: mark point A.
  • Point tool: mark point B.
  • Point tool: mark point D, off the line AB.
  • Segment tool: join A to B — one pair of opposite sides.
  • Parallel tool: click D, then line AB — a line through D parallel to AB (it carries side DC).
  • Compass tool: click A then B to open the compass exactly to the length AB.
  • Compass tool: keep the opening and click D — an arc of radius AB swings from D and crosses the parallel line.
  • Point tool: mark C where that arc meets the parallel line. Now DC = AB and DC ∥ AB.
  • Segment tool: join A to D.
  • Segment tool: join B to C — notice AD and BC come out equal and parallel too.

The proof, step by step

Prove that a quadrilateral with one pair of opposite sides equal and parallel is a parallelogram.

  • ∠1 = ∠2 (Alternate angles, since AB // DC and AC is transversal)
  • AB = DC (Given)
  • AC = AC (Common side)
  • ΔABC ≅ ΔCDA (By SAS Congruence)
  • ∠3 = ∠4 (CPCT - Corresponding Parts of Congruent Triangles)
  • AD // BC (If alternate angles are equal, lines are parallel)
  • ABCD is a parallelogram (Both pairs of opposite sides are parallel)

Worked example

In a quadrilateral ABCD, AB = DC and AB // DC. If ∠BAC = 35°, find the measure of ∠DCA.

Since AB // DC and AC is a transversal, alternate angles ∠BAC and ∠DCA are equal. Therefore, ∠DCA = ∠BAC = 35°.

  • 35° — correct
  • 45°
  • 55°
  • 70°
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