Chapter 5.1, Problem 15WE

To determine

To prove: Parallelogram $QM\cong MS\text{ and }TM\cong MR$ are corresponding parts of congruent triangles and diagonal of a parallelogram bisects each other.

Explanation of Solution

Given information:

  

QRST is a parallelogram.

Diagonal of a parallelogram bisects each other.

Explaination:

It is given that $QRST$ is a parallelogram.

The line $TS\parallel QR$ and $QS$ is a transversal.

Then the alternative interior angles are equal.

  $\angle 4\cong \angle 3$ ........... (1)

  $\angle 2\cong \angle 1$

  $\text{Δ}QMR$ and $\text{Δ}TMS$

  $\angle 4\cong \angle 3$ From (1)

  $\angle 2\cong \angle 1$

Opposite sides of the parallelogram are equal

Thus,

  $QR\cong TS$

Therefore, by angle side angle property,

  $△ADB\cong △BCD$

Thus, by corresponding parts of congruent triangles,

  $\begin{array}{l}QM\cong MS\hfill \\ TM\cong MR\hfill \end{array}$

We have given $ABCD$ is a parallelogram.

  $EF$

  $\parallel HG$ And EG is a transversal.

  $\angle 4\cong \angle 3$ (Alternate interior angle)

  $\angle 2\cong \angle 1$

In $\text{Δ}HEG$ and $\text{Δ}GFE$

  $\angle 4\cong \angle 3$ From (1)

  $EG\cong GE$ (Common side of a triangle)

  $\angle 2\cong \angle 1$ From (1)

  $△HEG\cong \text{Δ}GFE$ (Angle-Side-Angle Test)

  $EF\cong HG$

  $FG\cong EH$ (Corresponding sides of congruent ttriangles)

Therefore, bisection of diagonal of a parallelogram occurs at $FG\cong EH$ .