The Triangle Angle Sum Theorem states that the sum of the interior angles of a triangle is always 180 degrees.
For the triangle with angle measures \( (x - 20)^\circ \), \( (3x + 3)^\circ \), and \( (2x - 7)^\circ \), we set up the equation:
\[ (x - 20) + (3x + 3) + (2x - 7) = 180 \]
Now, combine like terms:
\[ x - 20 + 3x + 3 + 2x - 7 = 180 \]
This simplifies to:
\[ (1x + 3x + 2x) + (-20 + 3 - 7) = 180 \]
\[ 6x - 24 = 180 \]
Next, add 24 to both sides:
\[ 6x = 204 \]
Now divide by 6:
\[ x = 34 \]
Now we can find the measures of the angles:
- \( (x - 20)^\circ = (34 - 20)^\circ = 14^\circ \)
- \( (3x + 3)^\circ = (3 \cdot 34 + 3)^\circ = (102 + 3)^\circ = 105^\circ \)
- \( (2x - 7)^\circ = (2 \cdot 34 - 7)^\circ = (68 - 7)^\circ = 61^\circ \)
Now, we have the angle measures:
- \( 14^\circ \)
- \( 105^\circ \)
- \( 61^\circ \)
The largest angle is \( 105^\circ \).
Thus, the largest angle in the triangle is:
\[ \boxed{105} \]
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