To find the measures of the angles given the expressions for their measures:
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We know that vertical angles are equal, so we can set the measures equal to each other: \[ m∠2 = m∠4 \] This gives us the equation: \[ 5x + 23 = 3x + 35 \]
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Now, we'll solve for \( x \):
- Subtract \( 3x \) from both sides: \[ 5x - 3x + 23 = 35 \] \[ 2x + 23 = 35 \]
- Now, subtract 23 from both sides: \[ 2x = 12 \]
- Finally, divide by 2: \[ x = 6 \]
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Now that we have \( x \), we can substitute it back into either expression to find the measure of the angles. Let's use \( m∠2 \): \[ m∠2 = 5(6) + 23 \] \[ m∠2 = 30 + 23 = 53° \]
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Now let's check \( m∠4 \) for consistency: \[ m∠4 = 3(6) + 35 \] \[ m∠4 = 18 + 35 = 53° \]
Both angles are equal, confirming they measure \( 53° \).
So, the answer is: 53°