To solve for [tex]\(x\)[/tex], let's follow the reasoning and steps provided in the problem carefully. Here is the detailed step-by-step solution:
1. Given Information:
- We have three angles [tex]\(\hat{S}_1\)[/tex], [tex]\(\hat{S}_2\)[/tex], and [tex]\(\hat{S}_3\)[/tex].
- We know that [tex]\(\hat{S}_3 = x\)[/tex].
2. Assigning Values:
- It is given that [tex]\(\hat{S}_1 = x\)[/tex] and [tex]\(\hat{S}_2 = x\)[/tex].
- So, we have:
[tex]\[
\hat{S}_1 = x, \quad \hat{S}_2 = x, \quad \hat{S}_3 = x
\][/tex]
3. Sum of Angles:
- In any triangle, the sum of the interior angles is [tex]\(180^\circ\)[/tex]. Therefore:
[tex]\[
\hat{S}_1 + \hat{S}_2 + \hat{S}_3 = 180^\circ
\][/tex]
4. Substitute Values:
- Substitute the values of [tex]\(\hat{S}_1\)[/tex], [tex]\(\hat{S}_2\)[/tex], and [tex]\(\hat{S}_3\)[/tex] into the equation:
[tex]\[
x + x + x = 180^\circ
\][/tex]
5. Simplify the Equation:
- Combine like terms:
[tex]\[
3x = 180^\circ
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
- To isolate [tex]\(x\)[/tex], divide both sides of the equation by 3:
[tex]\[
x = \frac{180^\circ}{3}
\][/tex]
- Perform the division:
[tex]\[
x = 60^\circ
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] is [tex]\(60^\circ\)[/tex].
