Question

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

Answer 1

$\huge\star\underline{\mathtt\red{A}\mathtt\green{N}\mathtt\blue{S}\mathtt\purple{W}\mathtt\orange{E}\mathtt\pink{R}}\star\:$

ABCD is a rhombus.

⇒AC=BC ........(1)(given)

BC=AB .....(2)(side of rhombus)

From eqns(1) and (2)

AC=BC=AB

⇒△ABC is equilateral triangle.

So, ∠ABC=60

∘

∠BCA=60

∘

..........(3)

∠CAB=60

∘

..........(4)

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

AD=BC

But BC=AC

∴AD=AC

∴AD=DC=AC

∴DAC is an equilateral triangle.

⇒∠CAD=60

∘

......(5)

⇒∠ADC=60

∘

⇒∠DCA=60

∘

.......(6)

From eqns(3) and (6) we get

∠BCA+∠DCA=60

∘

+60

∘

=120

∘

∴∠C=120

∘

From eqns(4) and (5),

∠CAB+∠CAD=60

∘

+60

∘

=120

∘

∴∠A=120

∘

Hence the four angles of the Rhombus are 120

∘

,60

∘

,120

∘

,60

∘

Answer 2

Answer:

ABCD is a rhombus.

⇒AC=BC ........(1)(given)

BC=AB .....(2)(side of rhombus)

From eqns(1) and (2)

AC=BC=AB

⇒△ABC is equilateral triangle.

So, ∠ABC=60

∘

∠BCA=60

∘

..........(3)

∠CAB=60

∘

..........(4)

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

AD=BC

But BC=AC

∴AD=AC

∴AD=DC=AC

∴DAC is an equilateral triangle.

⇒∠CAD=60

∘

......(5)

⇒∠ADC=60

∘

⇒∠DCA=60

∘

.......(6)

From eqns(3) and (6) we get

∠BCA+∠DCA=60

∘

+60

∘

=120

∘

∴∠C=120

∘

From eqns(4) and (5),

∠CAB+∠CAD=60

∘

+60

∘

=120

∘

∴∠A=120

∘

Hence the four angles of the Rhombus are 120

∘

,60

∘

,120

∘

,60

∘