Question

Given:In quadrilateral WXYZ , `bar(WY)` and `bar(XZ)` bisect each other at point A.
Prove: Quadrilateral WXYZ is a parallelogram.

Answer 1

Consider the quadrilateral WXYZ, it is given that the diagonals bisect each other that means ZO = OX and WO = OY.

To prove: The quadrilateral WXYZ is a parallelogram.

Proof:

Consider the triangles ZOY and WOX,

Here, OZ = OX

WO = OY

$\angle ZOY = \angle WOX$ (Vertically opposite angles)

Therefore, $\Delta ZOY\cong \Delta WOX$ By SAS criteria

Therefore, ZY = WX and $\angle YZO = \angle OXW , \angle ZYO = \angle OWX$ (By Cpct)

Consider the triangles ZOW and YOX,

Here, OZ = OX

WO = OY

$\angle ZOW = \angle YOX$ (Vertically opposite angles)

Therefore, $\Delta ZOW\cong \Delta YOX$ By SAS criteria

Therefore, ZW = YX and $\angle WZO = \angle OXY , \angle ZWO = \angle OYX$ (By Cpct)

Therefore, now we get ZY=WX , ZW=YX that is opposite sides of the given quadrilateral are equal.

Since, $\angle YZO = \angle OXW , \angle ZYO = \angle OWX$

$\angle WZO = \angle OXY , \angle ZWO = \angle OYX$

Consider $\angle WZY = \angle WZO + \angle OZY$

$\angle WZY = \angle OXY + \angle OXW$

$\angle WZY = \angle WXY$

Now, consider  $\angle ZWX = \angle ZWO + \angle OWX$

$\angle WZY = \angle OYX + \angle ZYO$

$\angle WZY = \angle ZYX$

Hence, opposite angles of the given quadrilateral are equal.

Hence, the given quadrilateral is a parallelogram.

Answer 2

Answer: the answer is the last one

Step-by-step explanation:

Thats what i got on the assignment