Vertical angles are equal, so we can set the measures of the two angles equal to each other:
\[ m∠2 = m∠4 \]
Substituting the given expressions:
\[ 5x + 23 = 3x + 35 \]
Now, solve for \(x\):
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Subtract \(3x\) from both sides: \[ 5x - 3x + 23 = 35 \] This simplifies to: \[ 2x + 23 = 35 \]
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Subtract \(23\) from both sides: \[ 2x = 12 \]
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Divide by \(2\): \[ x = 6 \]
Now, substitute \(x\) back into either angle to find the measure. We can use \(m∠2\):
\[ m∠2 = 5x + 23 = 5(6) + 23 = 30 + 23 = 53° \]
Now check \(m∠4\) to be sure:
\[ m∠4 = 3x + 35 = 3(6) + 35 = 18 + 35 = 53° \]
Both angles measure \(53°\).
Thus, the measure of each angle is \( \boxed{53°} \).
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