Question

ABCD is a rhombus.Diagonal AC divides it into equilateral triangles.If angle ADC=60degree .Find angle ABC and angle DCB​

Answer 1

ABCD is a rhombus, and AC diagonal makes two equilateral Triangles ABC and ADC, and,

Angle ADC = 60°

We have to find out:---

1) Angle ABC=?

2)Angle DCB= ?

In Triangle ABC and Triangle ADC :---

1) AC common,

2) AB =AD,

3) BC = DC,

Therefore Triangles are congruent to each other,

So, Angle ABC=ADC=60

Angle BAC =DAC..........(a)

Therefore , Angle BAD = angle BAC + angle DAC.

Now ,angle BCA= Angle DAC,(b),

There fore , Angle DCB = angle BCA + angle DAC,

We know sum of 3 angles of a Triangle is always 180°,

In triangle ADC:-----

Angle ADC + angle DAC + angle DAC = 180°,

Angle DAC+ angle DCA=180°-60° ( angle = 60°),

Angle DAC +angle DCA= 120°,

Angle 2DAC= 120°,

Angle DAC =60°,

So angle DCA= 60°,

Same In triangleABC

We get,

Angle ABC+ angle ACB + CAB= 180°,

AngleACB+ angle CAB= 180° -60° = 120°

Angle 2 ACB= 120°

Angle ACB= 120°/2= 60°,

Therefore ,

Angle CAB = 60°,

Now Angle DCB = angle ACB + angle DCA,

Angle DCB = 60° +60°= 120°,

Ans:-- angleABC is 60°, and angle DCB is 120°,

Answer 2

∠DCB =120°

∠ABC=60°

Step-by-step explanation:

In  a rhombus ABCD,AC is a diagonal,which divides rhombus into two equilateral triangle ADC and ABC.

∠ADC=60° [Given]

We know that,

In equilateral ΔADC,

∠ADC+∠DCA+∠CAD=180°

Also,in an equilateral triangle all the three angles are equal.

∠ADC=∠DCA=∠CAD

⇒∠ADC=∠DCA=∠CAD=60°..............(1)

Also,opposite angle of a rhombus are always equal

⇒∠ADC=∠ABC

⇒∠ABC=60°

Now,In equilateral ΔABC,

∠ABC=∠ACB=∠BAC

⇒∠ABC=∠ACB=∠BAC=60°..............(2)

Now,∠DCB=∠DCA+∠BAC

                 =60°+60° [From equation (1) and (2)]

          ∠DCB =120°

           ∠ABC=60°