Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium]
234. Opposite Sides Equal · The parallelogram's hidden symmetry
Opposite sides of a parallelogram are always equal in length.
In a parallelogram, opposite sides are equal (AB = DC and AD = BC). A diagonal divides it into two congruent triangles (ASA), and CPCTC then makes the opposite sides equal.
What this lesson covers
Try to break it
Drag C to skew the parallelogram. AB still equals DC and AD still equals BC at every position. Try to find a shape where opposite sides come out unequal — impossible. Parallel sides force equal lengths.
How you build it
Construct a parallelogram with a diagonal.
- Drop vertex A at the lower left.
- Drop vertex B to the right of A.
- Connect A and B.
- Drop vertex C above the line AB, to the right.
- Connect B and C.
- Set the compass width to BC — click B, then C.
- Keep the width. Click A — the arc of radius BC is drawn (D lies on it, since AD = BC).
- Re-set the compass width to AB — click A, then B.
- Keep the width. Click C — the arc of radius AB is drawn (D lies on it too, since CD = AB).
- Click the crossing of the two arcs to mark D — the fourth vertex of the parallelogram.
- Connect C and D.
- Connect D and A.
- Connect A and C to draw the diagonal.
The proof, step by step
Prove that the opposite sides of a parallelogram are equal.
- In triangles ABC and CDA:
- ∠BAC = ∠DCA (Alternate angles, AB || DC)
- ∠BCA = ∠DAC (Alternate angles, AD || BC)
- AC = AC (Common side)
- ΔABC ≅ ΔCDA (ASA Congruence)
- AB = DC and AD = BC (CPCT)
Worked example
In a parallelogram ABCD, if AB = 8 cm and BC = 5 cm, what are the lengths of DC and AD respectively?
Opposite sides of a parallelogram are equal. Therefore, DC = AB = 8 cm and AD = BC = 5 cm.
- DC = 5 cm, AD = 8 cm
- DC = 8 cm, AD = 5 cm — correct
- DC = 8 cm, AD = 8 cm
- DC = 5 cm, AD = 5 cm