To solve the problem of finding the measure of angle [tex]\(C\)[/tex] in triangle [tex]\(ABC\)[/tex], given the measures of angles [tex]\(A\)[/tex] and [tex]\(B\)[/tex], let's follow these steps:
1. Recognize the given information:
- Angle [tex]\(A\)[/tex] measures [tex]\(90^\circ\)[/tex].
- Angle [tex]\(B\)[/tex] measures [tex]\(50^\circ\)[/tex].
2. Recall the key property of a triangle:
- The sum of the interior angles in any triangle is always [tex]\(180^\circ\)[/tex].
3. Set up an equation using this property:
- Let angle [tex]\(C\)[/tex] be the unknown angle.
- According to the angle sum property, we have:
[tex]\[
\angle A + \angle B + \angle C = 180^\circ
\][/tex]
4. Substitute the given angle measures into the equation:
- Substitute [tex]\( \angle A = 90^\circ \)[/tex] and [tex]\(\angle B = 50^\circ \)[/tex] into the equation, giving:
[tex]\[
90^\circ + 50^\circ + \angle C = 180^\circ
\][/tex]
5. Solve for [tex]\(\angle C\)[/tex]:
- Combine the known angle measures:
[tex]\[
140^\circ + \angle C = 180^\circ
\][/tex]
- Subtract [tex]\(140^\circ\)[/tex] from both sides to isolate [tex]\(\angle C\)[/tex]:
[tex]\[
\angle C = 180^\circ - 140^\circ
\][/tex]
- Simplify the expression:
[tex]\[
\angle C = 40^\circ
\][/tex]
Given these steps, the correct conclusion is that [tex]\(\angle C\)[/tex] must measure [tex]\(40^\circ\)[/tex].
Therefore, based on the possible answer choices:
- Angle [tex]\(C\)[/tex] must measure [tex]\(40^\circ\)[/tex] is the correct statement.
