To provide an answer to your question regarding the value of \( m \) that proves quadrilateral \( ABCD \) is a parallelogram, I would need to see the image you're referencing. However, I can offer some general guidance on how to approach problems involving parallelograms and their properties.
In a parallelogram, opposite sides are equal in length and parallel. If the diagonals intersect at point \( O \), then the line segments connecting the vertices to point \( O \) can provide information about the lengths of the sides.
For example, if you have lengths labeled from point \( O \) to the vertices and expressions involving \( m \), you would typically set up equations based on the equal lengths of the opposite sides (e.g., \( AO = OC \) and \( BO = OD \)) or you might be examining angles or other relationships involving the diagonals.
To find the specific value of \( m \) that ensures \( ABCD \) is a parallelogram, look for equalities that relate to the properties mentioned above.
If you can provide me the length expressions or relationships involving \( m \) that are shown in the image, I can help derive the correct value from there.