Question

In the adjoining figure, AC = BC and CD = AD. If ∠ADC = 60°. Find ∠BAC.

Answer 1

Given:

• B, C, D are collinear with C between B and D.
• AC = BC and CD = AD.
• ∠ADC = 60°.

Step-wise calculation:

1. In triangle ADC, we have AD = CD and ∠ADC = 60°.

Therefore, the two base angles at A and C are equal and 2 × (base angle) + 60° = 180°.

⇒ Each base angle = 60°. 

Hence, triangle ADC is equilateral.

So, AC = AD = CD.   ...(Isosceles/equilateral angle facts used)

2. From AC = AD = CD and the given AC = BC, we get

BC = AC = CD

So, BC = CD. 

Thus, C is the midpoint of BD (BC and CD are equal collinear segments).

3. At point C, the rays CB and CD are in opposite directions, 

So ∠ACB + ∠ACD = 180°.

Since ∠ACD = 60° from the equilateral triangle,

∠ACB

= 180° – 60°

= 120°

4. In triangle ABC, the sides AC and BC are equal. 

So, base angles at A and B are equal. 

Let ∠BAC = ∠ABC = x. 

Then x + x + 120° = 180°

⇒ 2x = 60°

⇒ x = 30°

⇒ ∠BAC = 30°.