To find the measures of the vertical angles, we can set the two expressions equal to each other, since vertical angles are congruent. So, we have:
\[ m∠2 = m∠4 \] \[ 5x + 23 = 3x + 35 \]
Now, we can solve for \(x\):
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Subtract \(3x\) from both sides: \[ 5x - 3x + 23 = 35 \]
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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 that we have the value of \(x\), we can substitute it back into either angle measure to find the measures of \(m∠2\) and \(m∠4\).
Calculating \(m∠2\): \[ m∠2 = 5(6) + 23 = 30 + 23 = 53° \]
Calculating \(m∠4\): \[ m∠4 = 3(6) + 35 = 18 + 35 = 53° \]
Thus, the measures of the angles are:
- \(m∠2 = 53°\)
- \(m∠4 = 53°\)
Therefore, the correct answer is \(53\) degrees.