Quadrilaterals
37. Parallelogram · opposite sides always stay parallel
Opposite sides AB & DC and AD & BC are always parallel.
A parallelogram is a quadrilateral whose opposite sides are parallel. It has two pairs of parallel sides — AB ∥ DC and AD ∥ BC. Drag any vertex; the parallel sides stay parallel by construction.
What this lesson covers
Try to break it
Try dragging the handles. Can you make it so the opposite sides are NOT parallel? Nope! It stays a parallelogram.
How you build it
Construct a parallelogram.
- Place point A — the first vertex of the parallelogram.
- Place point B.
- Place point C, clearly off the line AB so the parallelogram has real area.
- Draw side AB (Segment tool: click A, then B).
- Draw side BC (Segment tool: click B, then C).
- With the Parallel tool, click point C, then click on side AB — it draws a line through C parallel to AB.
- With the Parallel tool, click point A, then click on side BC — it draws a line through A parallel to BC.
- Mark point D where the two parallel lines cross — your click snaps onto the exact crossing. That completes parallelogram ABCD: side AB is parallel to CD, and BC is parallel to DA, by construction.
The proof, step by step
Prove that both pairs of opposite sides of the figure are parallel.
- Given: Quadrilateral ABCD has AB || DC and AD || BC.
- Definition: A quadrilateral with both pairs of opposite sides parallel is called a parallelogram.
- Conclusion: Therefore, ABCD is a parallelogram.
Worked example
In quadrilateral ABCD, AB is parallel to DC and AD is parallel to BC. Which of the following best describes ABCD?
By definition, a quadrilateral with both pairs of opposite sides parallel is a parallelogram. Since AB || DC and AD || BC, ABCD is a parallelogram.
- A trapezium
- A parallelogram — correct
- A kite
- A rectangle