Question

Geometry Question: Given segment EA is parallel segment DB, segment EA is congruent to segment DB, and B is the mid of segment AC; Prove: segment EB is parallel to segment DC (reference diagram in picture)

Answer 1

Construction: Join ED.

The corresponding diagram is given below,

According to the given problem,

`\begin{gathered} AE=BD \\ AE\parallel BD \end{gathered}`

Since a pair of opposite sides are parallel and equal, it can be claimed that quadrilateral ABDE is a parallelogram.

Then, as a property of any parallelogram, it can be argued that,

`\begin{gathered} AB=DE \\ AB\parallel DE \end{gathered}`

Given that B is the mid-point of AC,

`\begin{gathered} AB=BC \\ AB\parallel BC \end{gathered}`

Combining the above two results,

`\begin{gathered} BC=DE \\ BC\parallel DE \end{gathered}`

It follows that ABCD also forms a parallelogram.

Again using the property that opposite sides of a parallelogram are equal and parallel. It can be claimed that,

`\begin{gathered} EB=DC \\ EB\parallel DC \end{gathered}`

Hence proved that segment EB is parallel to segment DC,

`\vec{EB}=\vec{DC}`