If any triangle and any parallelogram are on the same base and between same parallels let us prove logically that the area of triangular region is half the area in the shape of parallelogram region.

Given.

If any triangle and parallelogram are on the same base and between the same parallel.

Formula used.

Area of parallelogram = Base × Perpendicular

Area of triangle = × Base × Height

⇒ Property of parallel lines

Perpendicular distance between 2 parallel is always same.

Draw a parallelogram ABCD between 2 parallel lines PQ and RS

Draw triangle with base CD and point E on common line AB

As we know that if both parallelogram and triangle lies on same 2 parallel lines

Perpendicular height of both will be same

Height of triangle(EF) = perpendicular of parallelogram (AG)

⇒ By Property of parallel lines

As both parallelogram and triangle possess same base CD

Which is also Base of parallelogram

∴ Base of parallelogram = Base of triangle = CD

Area of triangle = × Base × Height

× CD × EF

× CD × AG ∵ EF = AG

× [Perpendicular × Base of parallelogram]

× [Area of Parallelogram]

Hence proved;