Question

Prove: If a segment whose endpoints lie on opposite sides of a parallelogram passes though the midpoint of a diagonal, that segment is bisected by the diagonal.

Short answer

It is proved that segment is bisected by diagonal.

Step-by-step solution

Step 1. Draw a diagram with appropriate instruction.

Consider a parallelogram ABCD. $\overline{BD}$is a diagonal of that parallelogram.$\overline{EF}$ is the line segment that bisects the diagonal.

Step 2. Description of step.

As ABCD is a parallelogram and opposite sides of a parallelogram are equal. $\overline{BD}$is a transversal. Then alternate interior angles are congruent $\angle FDG\cong EBG$.

As $\overline{EF}$is the line segment which bisects the diagonal then, $\overline{DG}=\overline{GB}$.

Step 3. Description of step.

Consider $\Delta DGF$and $\Delta BGE$,$\angle DGF\cong \angle BGE$, $\overline{DG}=\overline{GB}$ and$\angle FDG\cong EBG$ then by ASA postulate $\Delta DGF\cong \Delta BGE$.

Step 4. Description of step.

As $\Delta DGF\cong \Delta BGE$then by corresponding parts of congruent triangles, $\overline{EG}\cong \overline{GF}$.

Hence it is proved that the segment is bisected by diagonal.