Question

One of the diagonals of a rhombus and its sides are equal. Find the angles of the rhombus.

Answer 1

$ABCD$ is a rhombus.

$\Rightarrow AC=BC$ ........$\left(1\right)$(given)

$BC=AB$ .....$\left(2\right)$(side of rhombus)

From eqns$\left(1\right)$ and $\left(2\right)$

$AC=BC=AB$

$\Rightarrow △ABC$ is equilateral triangle.

So, $\angle ABC={60}^{\circ }$

$\angle BCA={60}^{\circ }$ ..........$\left(3\right)$

$\angle CAB={60}^{\circ }$ ..........$\left(4\right)$

Similarly, in $△ADC,AD=DC$(sides of a rhombus)

$AD=BC$

But $BC=AC$

$\therefore AD=AC$

$\therefore AD=DC=AC$

$\therefore DAC$ is an equilateral triangle.

$\Rightarrow \angle CAD={60}^{\circ }$ ......$\left(5\right)$

$\Rightarrow \angle ADC={60}^{\circ }$

$\Rightarrow \angle DCA={60}^{\circ }$ .......$\left(6\right)$

From eqns$\left(3\right)$ and $\left(6\right)$ we get

$\angle BCA+\angle DCA={60}^{\circ }+{60}^{\circ }={120}^{\circ }$

$\therefore \angle C={120}^{\circ }$

From eqns$\left(4\right)$ and $\left(5\right)$,

$\angle CAB+\angle CAD={60}^{\circ }+{60}^{\circ }={120}^{\circ }$

$\therefore \angle A={120}^{\circ }$

Hence the four angles of the Rhombus are ${120}^{\circ },{60}^{\circ },{120}^{\circ },{60}^{\circ }$