To find the value of [tex]\( x \)[/tex], we start by utilizing the property that vertical angles are always equal. Given [tex]\( \angle E \)[/tex] and [tex]\( \angle F \)[/tex] are vertical angles, their measures must be the same.
Given:
[tex]\[ m \angle E = 5x + 10 \][/tex]
[tex]\[ m \angle F = 7x - 12 \][/tex]
Since [tex]\( m \angle E = m \angle F \)[/tex], we set the expressions for their measures equal to each other:
[tex]\[ 5x + 10 = 7x - 12 \][/tex]
Next, we solve for [tex]\( x \)[/tex] by isolating it on one side of the equation:
1. Subtract [tex]\( 5x \)[/tex] from both sides:
[tex]\[ 10 = 7x - 5x - 12 \][/tex]
[tex]\[ 10 = 2x - 12 \][/tex]
2. Add 12 to both sides:
[tex]\[ 10 + 12 = 2x \][/tex]
[tex]\[ 22 = 2x \][/tex]
3. Divide both sides by 2:
[tex]\[ x = \frac{22}{2} \][/tex]
[tex]\[ x = 11 \][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( 11 \)[/tex].
