# In a ΔABC, if ∠B = 60° and the ratio of two sides is a : c = 2 : √3 + 1, then ∠A= ____.

45°, 40°, 35°, 55°

**Solution:**

We will use the cosine law and sine law to calculate the value of ∠ A.

To find the length of side B we use the cosine law:

b^{2} = a^{2} + c^{2} - 2ac × cos B

Where a and c are the lengths of the sides and B is the angle between them.

b^{2} = 2^{2} + (√3 + 1)^{2} - 2 × (2) × (√3 + 1) × cos 60˚

b^{2} = 4 + 3 + 1 + 2√3 - 2(√3 + 1)

b^{2} = 8 + 2√3 - 2√3 - 2

b^{2} = 6

b = √6

To find ∠ A we use the sine law:

Given that two sides and the angle in between,

b / sinB = a / sinA

√6/ sin60° = 2 / sinA

(√2 ×√3)/ sin60° = 2 / sinA

(√2 ×√3)/(√3 / 2) = 2 / sinA

√2 sinA = 1

sin A = 1/√2

A = 45°

## In a ΔABC, if ∠B = 60° and the ratio of two sides is a : c = 2 : √3 + 1, then ∠A= ____.

**Summary:**

The measure of ∠A is 45˚ if in a ΔABC,∠B = 60° and the ratio of two sides is a : c = 2 :√3 + 1.