Maths - RoderickT - Parallelograms on same base between parallels are equal

Parallelograms on same base between parallels have equal areas

We will consider three variations which covers all possible combinations.

1. Diagonal of one is side of other.

Draw parallelogram ABCD with AB parallel to DC and parallelogram ACPD with AC parallel to DP

So

area ∇ ACD = 1/2 area parallelogram ABCD

Similarly it can be shown that

∇ ACD = ∇ PDC and area ∇ ACD = 1/2 area parallelogram ACPD

So it also follows that area of

Parallelogram ABCD and ACPD have equal areas

2. Consider parallelograms when they do not form above construct, then they will be one of the following types:

For both types

^RAB = ^ADC (Comp angles of parallelogram)

^PAD = ^QDS (Comp angles of parallelogram)

^BAP = 2 rt.angles - ^RAB - ^PAD (angles on straight line)

= 2 rt.angles - ^ADC - ^QDS

=^CDQ (angles on straight line)

BA = CD (Opp. sides of parallelogram)

PA = QD (Opp. sides of parallelogram)

So

Consider when BP<BC

Par-gram ABCD = ∇BAP + Trap.PCDA

= ∇CDQ + Trap.PCDA

= Par-gram PADQ

Consider when BP>BC

Par-gram ABCD = ∇BAP+Tri.ADR-Tri.CPR

= ∇CDQ+Tri.ADR-Tri.CPR

= Par-gram PADQ