Question

In quadrilateral ABCD, diagonals AC and BD bisect one another: Quadrilateral ABCD is shown with diagonals AC and BD intersecting at point P. What statement is used to prove that quadrilateral ABCD is a parallelogram?
A. Angles ABC and BCD are congruent.
B. Sides AB and BC are congruent.
C. Triangles BPA and DPC are congruent.
D. Triangles BCP and CDP are congruent.

Answer 1

B. A parallelogram has congruent sides.

Answer 2

Answer with explanation:

It is given that , quadrilateral A BC D, diagonals AC and B D bisect one another at point P.

In ΔAPB and ΔCPD

AP=PC

BP=PD

∠APB =∠CPD→Vertically opposite angles

ΔAPB ≅ ΔCPD→→[SAS]

→AB=CD⇒[CPCT]

→∠A BP=∠C DP⇒[C PCT]

Alternate interior angles are equal , so lines are parallel.

⇒AB║CD, and AB=CD

Similarly, we can prove ΔAPD ≅ ΔBPC, to prove AD║BC, and AD=BC.

⇒A Quadrilateral is a parallelogram , if one pair of opposite sides is equal and parallel.

Option C: The statement which is used to prove that quadrilateral ABCD is a parallelogram→→Triangles B PA and D PC are congruent.