State true or false:

If a diagonal of a parallelogram bisects an angle, then it will also bisect the other angle.

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True

False

answer is A.

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Detailed Solution

Given a statement, if a diagonal of a parallelogram bisects an angle, then it will also bisect the other angle.
We need to check if the given statement is true or false.
A parallelogram holds the property of opposite sides being parallel and equal.
Now, we will consider a parallelogram ABCD according to the given statement such AC is a diagonal as shown below,
Question Image

Now, we can say that the lines

A B / / C D

, and so the diagonal BD acts as a transversal.
Thus, by the property of alternate interior angles, we have,

∠ 2 = ∠ 3 ∠ 1 = ∠ 4

But it is also given that the diagonal bisects one of the angles, so, let us assume that the diagonal bisects

∠ A B C

.
So, we also have,

∠ 1 = ∠ 2

From the three equalities, we can have the following equalities,

∠ 1 = ∠ 2 = ∠ 3 = ∠ 4

So, we can say that the diagonal BD bisects

∠ A D C

as well.
Thus, we conclude that for any parallelogram, if a diagonal bisects one angle, then it will also bisect the other angle.
Hence, the statement is true.
Hence, the correct option is (1).

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