To solve for the measure of the third angle in a triangle, we use the fact that the sum of the interior angles of a triangle is always [tex]\(180^{\circ}\)[/tex].
Given:
- The first angle ([tex]\(\angle 1\)[/tex]) is [tex]\(32^{\circ}\)[/tex].
- The second angle ([tex]\(\angle 2\)[/tex]) is [tex]\(110^{\circ}\)[/tex].
We need to find the third angle ([tex]\(\angle 3\)[/tex]).
Here is a step-by-step process to find [tex]\(\angle 3\)[/tex]:
1. Start with the sum of the interior angles of a triangle:
[tex]\[
\angle 1 + \angle 2 + \angle 3 = 180^{\circ}
\][/tex]
2. Substitute the given values into the equation:
[tex]\[
32^{\circ} + 110^{\circ} + \angle 3 = 180^{\circ}
\][/tex]
3. Combine the known angles:
[tex]\[
142^{\circ} + \angle 3 = 180^{\circ}
\][/tex]
4. To isolate [tex]\(\angle 3\)[/tex], subtract [tex]\(142^{\circ}\)[/tex] from both sides of the equation:
[tex]\[
\angle 3 = 180^{\circ} - 142^{\circ}
\][/tex]
5. Perform the subtraction:
[tex]\[
\angle 3 = 38^{\circ}
\][/tex]
Thus, the measure of the third angle is [tex]\(38^{\circ}\)[/tex].
The correct answer is [tex]\( \boxed{38^{\circ}} \)[/tex] (option J).
