According to this theorem ” two triangles having the same base and lies between the same parallel lines are equal in area”

In the above figure;

ABC and PBC are the two triangles having same base BC.

Also, both the triangles lies between the same parallel lines.

**To Prove:**

Area ( ABC ) = Area ( PBC )

**Construction:**

Draw line CD parallel to AB

Draw line CR parallel to BP

**Proof:**

After the construction we now have two parallelogram ABCD and PBCR.

Note that both the parallelogram have the same base and lie between the same parallel lines.

So according to **parallelogram area theorem** we can write;

Area ( ABCD ) = Area ( PBCR ) —> eq (1)

Now consider triangle ABC and parallelogram ABCD.

Both the figures have same base and lie between same parallel lines.

According to **parallelogram – triangle theorem**, we can write:

Area (ABC) = 1/2 Area (ABCD)

2 * Area (ABC) = Area (ABCD) —> eq (2)

Similar is the case of triangle PBC and parallelogram PBCR

Area (PBC) = 1/2 Area (PBCR)

2 * Area (PBC) = Area (PBCR) —> eq (3)

Applying eq (2) and (3) on equation (1), we get;

Area ( ABCD ) = Area ( PBCR )

2 * Area (ABC) = 2 * Area (PBC)

Area (ABC) = Area (PBC)

Hence Proved