What is the measurement of the smallest angle for a triangle with side lengths 23, 18, and 30 units, rounded to the nearest tenth of a degree?

To find the measurement of the smallest angle in a triangle given its side lengths, we can use the Law of Cosines. This law relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula is:

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

Where ( c ) is the side opposite angle ( C ), and ( a ) and ( b ) are the other two sides of the triangle.

In this case, we have a triangle with side lengths of 23, 18, and 30 units. To identify the smallest angle, we should find the angle opposite the longest side (which is 30 units) first, as it will be the largest, and then proceed to check the angles opposite the other two sides.

1. Let's find angle ( A ) opposite side 23 using:

[ 23^2 = 18^2 + 30^2 - 2(18)(30) \cdot \cos(A) ]

Calculating this gives us:

[ 529 = 324 + 900 - 1080 \cdot \cos(A) ]