41. Trapezium · exactly one pair of parallel sides

Try to break it

Watch closely! If you drag the corners until AD becomes parallel to BC, the shape changes its identity. It's no longer a trapezium—it's a parallelogram! A trapezium must keep exactly one pair parallel.

How you build it

Construct a trapezium.

• Place point A — one end of the first base.
• Place point B. AB is the first base of the trapezium.
• Draw base AB (Segment tool: click A, then B).
• Place point C well above or below line AB — this gives the trapezium its height.
• With the Parallel tool, click point C, then click on base AB — it draws a line through C parallel to AB. The second base will lie along this line.
• Mark point D on the parallel line, on the side toward A — CD becomes the second base, parallel to AB.
• Draw leg BC (Segment tool: click B, then C).
• Draw leg DA (Segment tool: click D, then A). ABCD is a trapezium — bases AB and CD are parallel; legs BC and DA are not.

The proof, step by step

Prove that the trapezium has exactly one pair of parallel sides.

• AB and DC are parallel by construction (horizontal lines).
• AD and BC are not parallel. Their slopes differ as long as the horizontal shifts differ.
• ∠A + ∠D = 180° because they are co-interior angles between parallel lines AB and DC.
• ∠B + ∠C = 180° for the same reason. This confirms the trapezium angle property.

Worked example

In a trapezium ABCD, AB || DC. If ∠A = 110°, find ∠D.

Since AB || DC, ∠A and ∠D are co-interior angles. Their sum is 180°. So ∠D = 180° - 110° = 70°.

• 70° — correct
• 110°
• 90°
• 50°