Vertical angles are equal, meaning that the measures of angles \( m∠2 \) and \( m∠4 \) can be set equal to each other. Therefore, we have the equation:
\[ 5x + 23 = 3x + 35 \]
To solve for \( x \), we'll first move the terms involving \( x \) to one side and the constant terms to the other side. This can be done by subtracting \( 3x \) from both sides:
\[ 5x - 3x + 23 = 35 \]
This simplifies to:
\[ 2x + 23 = 35 \]
Next, we subtract 23 from both sides to isolate the term with \( x \):
\[ 2x = 35 - 23 \] \[ 2x = 12 \]
Now, divide both sides by 2 to solve for \( x \):
\[ x = 6 \]
Now that we have the value of \( x \), we can substitute it back into the expressions for \( m∠2 \) and \( m∠4 \) to find their measures.
First, substitute \( x = 6 \) into \( m∠2 \):
\[ m∠2 = 5x + 23 = 5(6) + 23 = 30 + 23 = 53^\circ \]
Next, substitute \( x = 6 \) into \( m∠4 \):
\[ m∠4 = 3x + 35 = 3(6) + 35 = 18 + 35 = 53^\circ \]
Thus, the measures of the angles are:
\[ m∠2 = 53^\circ \quad \text{and} \quad m∠4 = 53^\circ \]
Both angles measure \( 53^\circ \).
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