Question

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

Prove: Quadrilateral WXYZ is a parallelogram.

Answer 1

Answer: the answer is the last one

Explanation:

Thats what i got on the assignment

Answer 2

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.