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We want to find the value of $x$ and the measures of the angles for the given triangle.
To do this, we will use the interior angles theorem. This theorem tells us that the measures of the interior angles of a triangle must add to $180^\circ.$ Applying the theorem, we can create an equation by adding the given expressions and setting the sum equal to $180.$ \begin{aligned} {\color{#0000FF}{4x}}+{\color{#009600}{6x}}+{\color{#FF0000}{8x}}=180 \end{aligned} Let's solve the equation for $x.$
$4x+6x+6x=180$
$18x=180$
$x=10$
Now that we know the value of $x,$ we can substitute $x=\textcolor{darkviolet}{10}$ into the given expressions to find the angle measures of the triangle. \begin{aligned} {\color{#0000FF}{4x}} \quad\Rightarrow\quad &4(\textcolor{darkviolet}{10})&=40 \\ {\color{#009600}{6x}} \quad\Rightarrow\quad &6(\textcolor{darkviolet}{10})&=60\\ {\color{#FF0000}{8x}}\quad\Rightarrow\quad &8(\textcolor{darkviolet}{10})&=80 \end{aligned}