ABCD is a rhombus. If ∠BAC = 38°, find :

(i) ∠ACB

(ii) ∠DAC

(iii) ∠ADC.

(i) In a rhombus, all sides are equal.

⇒ AB = BC = CD = DA

And, opposite angles of a rhombus are equal.

∠BAC = ∠ACB

It is given that ∠BAC = 38°.

So, ∠ACB = ∠BAC = 38°.

Hence, the value of ∠ACB is 38°.

(ii) Since ABC is a triangle, the sum of angles in a triangle is 180°.

Therefore,

⇒ ∠ABC + ∠BAC + ∠ACB = 180°

⇒ ∠ABC + 38° + 38° = 180°

⇒ ∠ABC + 76° = 180°

⇒ ∠ABC = 180° - 76°

⇒ ∠ABC = 104°

Since opposite angles of a rhombus are equal:

⇒ ∠ABC = ∠ADC

⇒ ∠ADC = 104°

As AD = CD, we have:

∠DAC = ∠DCA

Now,

⇒ ∠DAC = $\dfrac{1}{2}$ [180° - 104°]

= $\dfrac{1}{2}$ [76°]

= 38°

Hence, the value of ∠DAC is 38°.

(iii) Since opposite angles of a rhombus are equal:

⇒ ∠ABC = ∠ADC

⇒ ∠ADC = 104°

Hence, the value of ∠ADC is 104°.