What is the measure of the smallest angle of the triangle whose sides have lengths of 4.3, 5.1 and 6.3?

 To determine the measure of the smallest angle in the given triangle, you can use the Law of Cosines. For a triangle with sides of length \(a\), \(b\), and \(c\), and corresponding angles \(A\), \(B\), and \(C\), the Law of Cosines states:

\[c^2 = a^2 + b^2 - 2ab \cos(C)\]

In this case, let \(a = 4.3\), \(b = 5.1\), and \(c = 6.3\). To find the smallest angle, you are interested in the angle opposite the smallest side. Let \(C\) be the angle opposite side \(c\).

\[6.3^2 = 4.3^2 + 5.1^2 - 2 \cdot 4.3 \cdot 5.1 \cos(C)\]

Solving for \(\cos(C)\):

\[39.69 = 18.49 + 26.01 - 43.53 \cos(C)\]

Combine like terms:

\[39.69 = 44.50 - 43.53 \cos(C)\]

Rearrange to isolate \(\cos(C)\):

\[-4.81 = -43.53 \cos(C)\]

Divide by \(-43.53\):

\[\cos(C) \approx 0.1106\]

Now, find the angle \(C\) by taking the arccosine:

\[C \approx \cos^{-1}(0.1106)\]

Using a calculator, you find:

\[C \approx 83.79^\circ\]

Therefore, the smallest angle in the triangle is approximately \(83.79^\circ\).