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Prove that a diagonal of a Parallelogram divides it into two congruent triangles.
A diagonal of a Parallelogram divides it into two congruent triangles.
To do :
We have to prove that a diagonal of a Parallelogram divides it into two congruent triangles.
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.
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