To find the measure of [tex]\(\angle J\)[/tex], the smallest angle in a triangle with sides measuring 11, 13, and 19, we need to use the Law of Cosines. The Law of Cosines states:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]
where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the sides of the triangle, and [tex]\(A\)[/tex] is the angle opposite side [tex]\(a\)[/tex].
First, we'll determine each angle of the triangle using the given side lengths.
1. Finding [tex]\(\angle A\)[/tex] (opposite to side [tex]\(a = 11\)[/tex]):
[tex]\[
\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}
\][/tex]
Plugging in the values:
[tex]\[
\cos(A) = \frac{13^2 + 19^2 - 11^2}{2 \cdot 13 \cdot 19}
\][/tex]
Upon calculating, we find:
[tex]\[
\cos(A) \approx 0.829803
\][/tex]
Therefore,
[tex]\[
\angle A \approx \cos^{-1}(0.829803) \approx 34.11^\circ
\][/tex]
2. Finding [tex]\(\angle B\)[/tex] (opposite to side [tex]\(b = 13\)[/tex]):
[tex]\[
\cos(B) = \frac{a^2 + c^2 - b^2}{2ac}
\][/tex]
Plugging in the values:
[tex]\[
\cos(B) = \frac{11^2 + 19^2 - 13^2}{2 \cdot 11 \cdot 19}
\][/tex]
Upon calculating, we find:
[tex]\[
\cos(B) \approx 0.749052
\][/tex]
Therefore,
[tex]\[
\angle B \approx \cos^{-1}(0.749052) \approx 41.51^\circ
\][/tex]
3. Finding [tex]\(\angle C\)[/tex] (opposite to side [tex]\(c = 19\)[/tex]):
[tex]\[
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
\][/tex]
Plugging in the values:
[tex]\[
\cos(C) = \frac{11^2 + 13^2 - 19^2}{2 \cdot 11 \cdot 13}
\][/tex]
Upon calculating, we find:
[tex]\[
\cos(C) \approx -0.248855
\][/tex]
Therefore,
[tex]\[
\angle C \approx \cos^{-1}(-0.248855) \approx 104.37^\circ
\][/tex]
After calculating the three angles, we identify the smallest angle among [tex]\(\angle A\)[/tex], [tex]\(\angle B\)[/tex], and [tex]\(\angle C\)[/tex]. The angles are approximately [tex]\(34.11^\circ\)[/tex], [tex]\(41.51^\circ\)[/tex], and [tex]\(104.37^\circ\)[/tex], respectively.
Thus, the smallest angle is [tex]\(\angle A \approx 34.11^\circ\)[/tex].
Rounding to the nearest whole degree, the smallest angle [tex]\(\angle J\)[/tex] is:
[tex]\[
\boxed{34^\circ}
\][/tex]
