3. Prove that parallelogram is a rectangle if its diagonals are of equal lengths.
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3. Prove that parallelogram is a rectangle if its diagonals are of equal lengths.
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Answer:
Let's draw a figure (See file attached below, I used ms paint)
Remember, I drew a rectangle because we have to prove that it is an rectangle.
Given: ABCD is a parallelogram with AC=BD.
Now,
In △ABC and △ABD
⇒ △ABC ≅ △BAD [ by SSS congruence]
Then, by CPCT, ∠ABC = △BAD
Now, ∠ABC + ∠BAD = 180°----(2) [interior angles on the same side of the transversal are sypplementary]
But, ∠ABC = △BAD -----(1)
From (1) and (2), we get:
∠ABC + ∠ABC = 180° [∵ ∠ABC = ∠BAD]
⇒ 2∠ABC = 180°
⇒ ∠ABC = 180°÷2
⇒ ∠ABC = 90°
Now, we know that If one of the interior angles of a parallelogram is 90°, is is an rectangle.
Since ABCD is a parallelogram and one of its interior angles is 90º, ABCD is a rectangle.
HOPE THIS HELPS :D