Calculate the value of [tex]$x$[/tex].

[tex]\[
\begin{tabular}{|c|c|}
\hline
\hat{S}_1 + \hat{S}_2 + \hat{S}_3 = & REASON \\
\hline
\hat{S}_3 = x & \\
\hline
\begin{tabular}{l}
then: \\
\hat{S}_2 = \\
\qquad \\
x and \\
\hat{S}_1 = \\
\qquad \\
x
\end{tabular} & Given \\
\hline
\begin{tabular}{l}
\therefore x + x + x = 180^{\circ} \\
x =
\end{tabular} & Substitute $\hat{S}_1, \hat{S}_2,$ and $\hat{S}_3$ \\
\hline
\end{tabular}
\][/tex]

1.3



Answer :

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].