Question

Prove the following statement.
Opposite angles of a parallelogram are equal.

Answer 1

Hint: Let us draw a parallelogram ABCD and use the result that diagonals are transversal to opposite sides of the diagonal because opposite sides of a parallelogram are parallel to each other.

Complete Step-by-Step solution:

As we know that any figure having four sides is known as quadrilateral.
And the quadrilateral whose opposite sides are parallel to each other is known as parallelogram.
So, Let the figure drawn ABCD be a parallelogram.

 

Now as we know that sides AB is parallel to DC (AB || DC).
And sides AD and BC are parallel (AD || BC).
As we know if two lines are parallel to each other then the line drawn from one side of the one parallel to the other side of the other parallel line is known as transversal.

Like if WX and YZ are the two parallel lines then XY will be the transversal to the lines WX and YZ.
And due to the alternate angle property \[\angle WXY = \angle ZYX\].
So, in the parallelogram ABCD
BD will be the transversal to lines AB and Dc.
\[\angle ABD = \angle CDB\] (1)
\[\angle CBD = \angle ADB\] (2)
Now adding equation 1 and equation 2. We get,
\[\angle B = \angle D\]
Similarly, \[\angle A = \angle C\]
And angle B and angle D are opposite to each other. And angles A and C are also opposite to each other.
Hence, the opposite angles of the parallelogram are equal.

Note: Whenever we come up with this type of the problem then first, we had to draw a parallelogram ABCD and the remember that if two lines WX and YZ are parallel to each other than the line from one side of the one line to the other side of the other line is transversal (i.e. XY). And from the property of angles angle to the opposite sides of the transversal are equal (i.e. \[\angle WXY = \angle ZYX\]). After using this identity, we will get the required result.