To solve the problem, we start by using the Triangle Angle Sum Theorem, which states that the sum of the interior angles of a triangle is 180 degrees.
The angles of the triangle are given as:
- \( 2x + 6 \)
- \( 4x - 1 \)
- \( 6x + 7 \)
Adding these angles together, we get:
\[ (2x + 6) + (4x - 1) + (6x + 7) = 180 \]
Simplifying the left side:
\[ 2x + 6 + 4x - 1 + 6x + 7 = 180 \]
Combine like terms:
\[ (2x + 4x + 6x) + (6 - 1 + 7) = 180 \]
This reduces to:
\[ 12x + 12 = 180 \]
Next, we solve for \( x \) by isolating it:
\[ 12x = 180 - 12 \] \[ 12x = 168 \] \[ x = \frac{168}{12} = 14 \]
Now that we have \( x = 14 \), we can find the measures of the three angles:
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First angle: \[ 2x + 6 = 2(14) + 6 = 28 + 6 = 34 \]
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Second angle: \[ 4x - 1 = 4(14) - 1 = 56 - 1 = 55 \]
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Third angle: \[ 6x + 7 = 6(14) + 7 = 84 + 7 = 91 \]
The angles are:
- \( 34^\circ \)
- \( 55^\circ \)
- \( 91^\circ \)
The smallest angle is \( 34^\circ \). The exterior angle corresponding to this smallest angle can be found using the property that the exterior angle is equal to the sum of the two opposite interior angles.
Thus, the exterior angle is: \[ 55 + 91 = 146^\circ \]
Therefore, the measure of the angle exterior to the smallest angle of the triangle is \( \boxed{146} \).
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