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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.

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parmesanchilliwack parmesanchilliwack · Answer 1 · 2018-11-22T13:26:53+01:00

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

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, [tex]\triangle BPA[/tex] and [tex]\triangle DPC[/tex].

Here, AP=PC

BP=PD,

[tex]\angle APB=\angle DPC[/tex] ( vertically opposite angles.)

So, By SAS postulate,[tex]\triangle BPA\cong \triangle DPC[/tex]

Thus AB=CD ( CPCT).

Similarly, we can prove, [tex]\triangle APD\cong \triangle BPC[/tex]

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.