To find the largest angle in the triangle with the measures \((x-20)^\circ\), \((3x+3)^\circ\), and \((2x-7)^\circ\), we use the Triangle Angle Sum Theorem, which states that the sum of the angles in a triangle is always 180 degrees. Therefore, we set up the equation:
\[ (x - 20) + (3x + 3) + (2x - 7) = 180 \]
Now, we can simplify the left side:
\[ x - 20 + 3x + 3 + 2x - 7 = 180 \] Combining like terms:
\[ (1x + 3x + 2x) + (-20 + 3 - 7) = 180 \] \[ 6x - 24 = 180 \]
Next, we solve for \(x\):
\[ 6x = 180 + 24 \] \[ 6x = 204 \] \[ x = 34 \]
Now that we have the value of \(x\), we can find the measures of each angle:
- \( (x - 20)^\circ = (34 - 20)^\circ = 14^\circ \)
- \( (3x + 3)^\circ = (3 \cdot 34 + 3)^\circ = (102 + 3)^\circ = 105^\circ \)
- \( (2x - 7)^\circ = (2 \cdot 34 - 7)^\circ = (68 - 7)^\circ = 61^\circ \)
Now we have the angle measures:
- \( 14^\circ \)
- \( 105^\circ \)
- \( 61^\circ \)
The largest angle in the triangle is:
\[ \boxed{105^\circ} \]
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