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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.
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