Question

NEED ASAP!!!!

In quadrilateral ABCD, diagonals AC and BD bisect one another:



What statement is used to prove that quadrilateral ABCD is a parallelogram?
Angles ABC and BCD are congruent.
Sides AB and BC are congruent.
Triangles BPA and DPC are congruent.
Triangles BCP and CDP are congruent.

Answer 1

Answer:Triangles BPA and DPC are congruent is used to prove that ABCD is a parallelogram.

Step-by-step explanation:Here, we have given a quadrilateral ABCD in which diagonals AC and BD bisect each other.

If P is a an intersection point of these diagonals

Then we can say that, AP=PC and BP=PD ( by the property of bisecting)

So, In quadrilateral ABCD,

Let us take two triangles,
`\triangle BPA` and
`\triangle DPC`.

Here, AP=PC

BP=PD,

`\angle APB=\angle DPC` ( vertically opposite angles.)

So, By SAS postulate,
`\triangle BPA\cong \triangle DPC`

Thus AB=CD ( CPCT).

Similarly, we can prove,
`\triangle APD\cong \triangle BPC`

Thus, AD=BC (CPCT).

Similarly, we can get the pair of congruent opposite angle for this quadrilateral ABCD.

Therefore, quadrilateral ABCD is a parallelogram.

Note: With help of other options we can not prove quadrilateral ABCD is a parallelogram.