To find the value of \( x \) given that the measures of the angles in a triangle are \( (2x)^\circ \), \( (3x)^\circ \), and \( (x + 60)^\circ \), follow these steps:
1. Recall the Triangle Sum Theorem: The sum of the angles in any triangle is \( 180^\circ \). Therefore, the sum of these given angles must equal \( 180^\circ \):
[tex]\[
(2x) + (3x) + (x + 60) = 180.
\][/tex]
2. Combine Like Terms: Add up all the \( x \) terms on the left side of the equation:
- The \( x \) terms are \( 2x \), \( 3x \), and \( x \).
[tex]\[
2x + 3x + x = 6x.
\][/tex]
- Now, the equation becomes:
[tex]\[
6x + 60 = 180.
\][/tex]
3. Isolate the \( x \) Term: First, subtract \( 60 \) from both sides of the equation to move the constant term to the right side:
[tex]\[
6x + 60 - 60 = 180 - 60,
\][/tex]
which simplifies to:
[tex]\[
6x = 120.
\][/tex]
4. Solve for \( x \): Divide both sides of the equation by \( 6 \) to solve for \( x \):
[tex]\[
x = \frac{120}{6} = 20.
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( 20 \)[/tex].
