Question

What is the measure of ∠W, rounded to the nearest degree?

19°
32°
56°
71°

Answer 1

71°

Explanation:

This is an isosceles triangle because it has two sides with lengths, hence the angles opposite the equal sides are also equal, that is ∠U = ∠V.

So we can say that:

∠U = ∠V = α

∠W = β

Since the internal angles of a triangle add up to 180 degrees, then:

α + α + β = 180

2α + β = 180

β = 180 - 2α

Using the law of sine:

`(35)/(sin\beta) =(30)/(sin\alpha) \\ \\ (35)/(sin(180 - 2\alpha)) =(30)/(sin\alpha) \\ \\ \\ From \ Properties: \\ \\ sin(180-2\alpha)=sin(180)cos2\alpha-sin2\alpha cos(180) \\ \\ = -sin2\alpha(-1)=sin2\alpha \\ \\ Also: \\ \\ sin2\alpha=2sin\alpha cos\alpha`

Therefore:

`(35)/(2sin\alpha cos\alpha) =(30)/(sin\alpha) \\ \\ \therefore (35)/(60)=cos\alpha \\ \\ \alpha=cos^(-1)((7)/(12))=54.31^(\circ)`

But we want to know ∠W = β, therefore:

`\beta = 180 - 2\alpha \\ \\ \beta =180-2(54.31)=71.37^(\circ)`

And rounded to the nearest degree:

∠W = 71°