Question

prove that the straight lines joining the midpoints of the opposite sides of a parallelogram are parallel to the other pairs of The parallel sides

Answer 1

use geometry book of 9 and 10 then use theorems

Answer 2

In the parallelogram ABCD, AB is parallel to CD and AC is parallel to BD.

E and F are mid points of AC and BD respectively.

AG is the perpendicular dropped from A to CD and

BH is the perpendicular dropped from B to CD ,

Since AB is parallel to CD, AG = BH,

( if and only if AG = BH, AB will be parallel to CD)

Similarly, AC = BD. (Only if AC = BD, AB will be parallel to CD).

Further, AC is parallel to BD and AG is parallel to BH.

In conclusion,

AC is parallel to BD and are equal to each other.

AG is parallel to BH and are equal to each other.

Triangles ACG and BDH are congruent.

Since E is mid point of AC and F is that of BD,

Triangles AEE’ and BFF’ are congruent.

AE and BF are equal and parallel

AE’ and BF’ are equal and parallel.

Hence AB is parallel to EF.

Since CD is parallel to AB,

AB, EF and CD are all parallel to each other.