Question

The four angles of a quadrilateral are equal. Draw this quadrilateral in your notebook. Find each of them.

Answer 1

Hint: A quadrilateral is a 2-dimensional geometrical shape which has four sides. Sum of interior angles of a polygon which has n sides is given by \[(n-2)\times {{180}^{\circ }}\], with this formula you can find the total sum of interior angles of the quadrilateral. All the four angles are given equal so suppose the measure of each angle be \[x\].

Complete step-by-step answer:
Given, all the four angles of the quadrilateral are equal. Let us draw a quadrilateral ABCD which has sides AB, BC, CD and DA.
 

The above figure is a quadrilateral ABCD. \[AB\], \[BC\], \[CD\] and \[DA\] are the sides of quadrilateral ABCD and \[\angle A\], \[\angle B\], \[\angle C\] and \[\angle D\] are the four angles of quadrilateral.
As it is given all the angles of quadrilateral ABCD are equal, let each angle be \[x\].
According to the formula for sum of all interior angles of a polygon, we have
Sum of all interior angles = \[(n-2)\times {{180}^{\circ }}\] , where n is the sides of a polygon.
Here in quadrilateral n = 4.
Sum of interior angles of quadrilateral = \[(4-2)\times {{180}^{\circ }}\]
= \[2\times {{180}^{\circ }}\]
= \[{{360}^{\circ }}\]
Now, sum of interior angle of quadrilateral = \[\angle A+\angle B+\angle C+\angle D\]
We can write is as
\[\angle A+\angle B+\angle C+\angle D={{360}^{\circ }}\]
As we know, \[\angle A=\angle B=\angle C=\angle D=x\]
We have,
\[x+x+x+x={{360}^{\circ }}\]
\[4x={{360}^{\circ }}\]
Hence \[x={{90}^{\circ }}\]
We have calculated the measure of each angle of quadrilateral ABCD which is equal to \[{{90}^{\circ }}\]
We can write it as \[\angle A=\angle B=\angle C=\angle D=x={{90}^{\circ }}\].

Note: In case we were given a polygon with more than four sides sum of interior angle can be obtained if we remember the formula to calculate the sum of interior angle \[(n-2)\times {{180}^{\circ }}\] where n is the side of a polygon. We can also try to recollect the common quadrilaterals – square, rectangle, parallelogram, rhombus and kite. Out of these, only square and rectangle have all interior angles as equal and the measure of each one is \[{{90}^{\circ }}\]. This will also help us answer this question.