Question

If the length of one side of a rhombus is equal to the length of one diagonal, find the angles of the rhombus.

Answer 1

Hint: If we have the rhombus as one side is equal to the diagonal then we have the rhombus divided into two equilateral triangles. So, other angles are equal as that is a rhombus and two adjacent angles will give us sum as \[180^\circ \] .

Complete step-by-step answer:
One of the diagonals of a rhombus is equal to one of its sides.
We know that a rhombus has four equal sides. If one side has the same length as a diagonal, the diagonal is part of two equilateral triangles. Since a rhombus has four sides we can say ,
Rhombus ABCD where AB \[ = \] diagonal of AC .

Since the triangle ABC is equilateral and so we have \[\angle ABC{\text{ }} = {\text{ }}60^\circ \] , \[\angle BCA{\text{ }} = {\text{ }}60^\circ \] , and \[\angle CAB{\text{ }} = {\text{ }}60^\circ \].
Also the triangle ADC is an equilateral triangle so we have \[\angle ADC{\text{ }} = {\text{ }}60^\circ \] , \[\angle DCA{\text{ }} = {\text{ }}60^\circ \] and \[\angle CAD{\text{ }} = {\text{ }}60^\circ \] .
Now, as we have \[\angle DAB{\text{ = }}\angle DAC + \angle CAB\]
On substituting the values of \[\angle DAC\] and \[\angle CAB\], we get,
 \[ \Rightarrow \angle DAB = {\text{ }}60^\circ + {\text{ }}60^\circ \]
On solving we get,
 \[ \Rightarrow \angle DAB = {\text{ 12}}0^\circ \]
Similarly we have, \[\angle DCB{\text{ = }}\angle DCA + \angle ACB\]
On substituting the values of \[\angle DCA\] and \[\angle ACB\], we get,
 \[ \Rightarrow \angle DCB = {\text{ }}60^\circ + {\text{ }}60^\circ \]
On solving we get,
 \[ \Rightarrow \angle DCB = {\text{ 12}}0^\circ \]
So, we have the angles as, \[\angle ABC{\text{ }} = {\text{ }}60^\circ = \angle ADC\] and \[\angle DCB = {\text{ 12}}0^\circ = \angle DAB\].

Note: Properties of Rhombus
1.All sides of the rhombus are equal.
2.The opposite sides of a rhombus are parallel.
3.Opposite angles of a rhombus are equal.
4.In a rhombus, diagonals bisect each other at right angles.
5.Diagonals bisect the angles of a rhombus.
6.The sum of two adjacent angles is equal to 180 degrees.