Prove that, in a parallelogram
1)opposite sides are equal
2) opposite angles are equal
3) Each diagonal will divide the parallelogram into two congruent triangles

Given :

We have to prove that , in a parallelogram, 

1)opposite sides are equal

2) opposite angles are equal

3) Each diagonal will divide the parallelogram into two congruent triangles

 

Solution :

Take a parallelogram ABCD and join

Two of its non-adjacent vertices, say A and C.

In the parallelogram ABCD,

BC || AD and AB || DC.

AC is the transversal between the parallel lines BC and AD, and it is also a transversal between the parallel lines AB and DC.

Let the angles formed by these parallel lines and transversal be angles 1, 2, 3 and 4.

Consider ∆ ABC and ∆ CDA,

∠ 1 = ∠ 3         (since alternate angles are equal)

∠ 2 = ∠ 4         (since alternate angles are equal)

AC = CA          (common side)

So, by the ASA rule ∆ ABC ≅ ∆ CDA.

Therefore, diagonal AC divides

parallelogram ABCD into two congruent triangles ABC and CDA.

 

In congruent triangles ABC and CDA,

BC = DA     (corresponding sides of congruent triangles)

Similarly, BA = DC.

BC and DA; BA and DC are opposite sides in the parallelogram ABCD.

Therefore, in a parallelogram, opposite sides are equal.

Hence proved.