253. Same Base, Same Parallels · Why two parallelograms always share the same area
Area of //gm ABCD always equals Area of //gm ABEF.
What this lesson covers
Try to break it
Drag H to change the distance between the two parallel lines (both parallelograms grow taller together). Drag D and F to shear each parallelogram independently along the top line. The bases are equal and the heights are equal, so the two areas always stay equal — try to give one a bigger area than the other, impossible.
How you build it
Construct two parallelograms on the same base.
- Point tool: mark A — the left end of the common base.
- Point tool: mark B — the right end of the common base.
- Segment tool: join A to B — the common base.
- Parallel tool: click a spot above, then click AB — the upper parallel that both parallelograms reach.
- Point tool: mark point D on the upper parallel — the top-left corner of the first parallelogram.
- Segment tool: join A to D — the left side.
- Parallel tool: click B, then click on AD — this is side BC; it meets the upper parallel at C.
- Point tool: mark point C where the line through B meets the upper parallel — ABCD is now a parallelogram.
- Point tool: mark point F somewhere else on the upper parallel — top-left of the second parallelogram.
- Segment tool: join A to F — the left side of the second parallelogram.
- Parallel tool: click B, then click on AF — this is side BE; it meets the upper parallel at E.
- Point tool: mark point E where the line through B meets the upper parallel. ABEF is now a parallelogram on the same base — ABCD and ABEF sit on base AB between the same two parallels, so they have equal area.
The proof, step by step
Prove that parallelograms on the same base and between the same parallels are equal in area.
- AD = BC (Opposite sides of //gm ABCD are equal)
- ∠ADF = ∠BCE (Corresponding angles, AB || DE)
- ∠AFD = ∠BEC (Corresponding angles, AB || DE)
- ∴ ΔADF ≅ ΔBCE [ASA Congruence Rule]
- Area(ΔADF) = Area(ΔBCE) (Congruent triangles have equal area)
- Area(ABCD) = Area(ABEF) [Adding Area(ABEF) to both sides]
Worked example
Two parallelograms ABCD and ABEF share the same base AB = 10 cm and lie between the same parallels. If the area of parallelogram ABCD is 80 cm², what is the area of parallelogram ABEF?
By Theorem 19, parallelograms on the same base and between the same parallels have equal areas. Hence, Area(ABEF) = Area(ABCD) = 80 cm².
- 40 cm²
- 80 cm² — correct
- 160 cm²
- Cannot be determined