# In triangle ABC, a = 1, b = 1, and C = 120°. Find the value of c.

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In triangle ABC, a = b implies angle A = angle B

Let A = B = x

The 3 angles of a triangle add up to 180.

x + x + 120 = 180

=> 2x = 60

=> x = 30

A = B = 30 degrees

Use the law of sines

a/sin A = b/sin B = c/sin C

=> 1/sin 30 = c/sin 120

=> c = sin 120/sin 30

=> c = sqrt 3

**The value of c = sqrt 3**

Since we have the length of two adjacent sides and the angle between them, we'll apply the law of cosines:

c^2 = a^2 + b^2 - 2a*b*cos C

c^2 = 1 + 1 - 2*1*1*cos (120)

But cos 120 = -cos 60 = -1/2

c^2 = 2 + 2*(1/2)

c^2 = 2+1

c^2 = 3

c = +sqrt3

Since the length of a side cannot be negative, we'll reject the negative value -sqrt3.

**The requested value of the length of the side c is: c = sqrt3.**