Question

given ( m parallel n ), find the value of ( x ).
( (3x - 23)^circ )
( (7x - 7)^circ )
answer attempt 1 out of 2
( x = )
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Explanation:

Step1: Identify angle relationship

Since \( m \parallel n \), the two angles \( (3x + 23)^\circ \) and \( (7x - 7)^\circ \) are alternate exterior angles, so they are equal.
\( 3x + 23 = 7x - 7 \)

Step2: Solve for \( x \)

Subtract \( 3x \) from both sides:
\( 23 = 4x - 7 \)

Add 7 to both sides:
\( 30 = 4x \)

Divide both sides by 4:
\( x = \frac{30}{4} = 7.5 \)? Wait, no, wait. Wait, maybe they are same - side? Wait, no, let's re - check. Wait, maybe they are supplementary? Wait, no, the diagram: if the transversal cuts the two parallel lines, and the angles are on the same side? Wait, no, maybe I made a mistake. Wait, let's look again. Wait, the angle \( (3x + 23)^\circ \) and \( (7x - 7)^\circ \): if \( m \parallel n \), and the transversal, maybe they are same - side interior angles? No, wait, maybe they are alternate exterior. Wait, no, let's calculate again. Wait, maybe I messed up the relationship. Wait, let's do the equation again.

Wait, if \( 3x + 23 = 7x - 7 \)

Then, \( 23 + 7=7x - 3x \)

\( 30 = 4x \)

\( x=\frac{30}{4}=7.5 \)? That seems odd. Wait, maybe the angles are supplementary? Let's check. If \( m \parallel n \), and the angles are same - side exterior angles, they are supplementary. So \( (3x + 23)+(7x - 7)=180 \)

Step3: Correct angle relationship (supplementary)

Combine like terms:
\( 3x+7x + 23 - 7=180 \)

\( 10x+16 = 180 \)

Subtract 16 from both sides:
\( 10x=180 - 16=164 \)

\( x = 16.4 \)? No, that's not right. Wait, maybe I misread the angles. Wait, the first angle is \( (3x + 23)^\circ \), the second is \( (7x - 7)^\circ \). Wait, maybe they are corresponding angles? Wait, no, let's look at the diagram again. The two lines are parallel, \( m \) and \( n \), and the transversal. The angle above \( m \) and the angle below \( n \): maybe they are vertical angles? No, vertical angles are equal. Wait, maybe the angles are equal. Wait, let's do the first equation again.

\( 3x + 23=7x - 7 \)

\( 23 + 7=7x - 3x \)

\( 30 = 4x \)

\( x = 7.5 \). But that seems like a decimal. Wait, maybe the problem has a typo? Or maybe I made a mistake. Wait, let's check the calculation again.

Wait, \( 3x+23 = 7x - 7 \)

Subtract \( 3x \): \( 23=4x - 7 \)

Add 7: \( 30 = 4x \)

\( x=\frac{30}{4}=7.5 \). So \( x = 7.5 \) or \( \frac{15}{2} \).

Wait, maybe the angles are supplementary. Let's try that. \( (3x + 23)+(7x - 7)=180 \)

\( 10x + 16 = 180 \)

\( 10x=164 \)

\( x = 16.4 \). No, that's not. Wait, maybe the angle \( (3x + 23) \) and \( (7x - 7) \) are same - side interior angles? Wait, same - side interior angles are supplementary. Wait, but the positions: if \( m \parallel n \), and the transversal, the angle above \( m \) and the angle below \( n \): maybe they are alternate exterior angles, which are equal. So the first calculation is correct. So \( x = 7.5 \). Wait, but maybe the problem is written as \( (3x + 23) \) and \( (7x - 7) \) are same - side exterior angles, which are supplementary. Wait, let's check with \( x = 7.5 \):

\( 3x+23=3\times7.5 + 23=22.5+23 = 45.5 \)

\( 7x - 7=7\times7.5-7 = 52.5 - 7=45.5 \). Oh! Wait, they are equal. So my first calculation was correct. So \( x = 7.5 \). Wait, but 7.5 is \( \frac{15}{2} \). So the value of \( x \) is 7.5.

Answer:

\( x = 7.5 \) (or \( \frac{15}{2} \))