Question

given ( m parallel n ), find the value of x. answer attempt 1 out of 2 ( x = ) submit answer

Explanation:

Step1: Identify angle relationship

Since \( m \parallel n \), the two angles \( (x + 5)^\circ \) and \( (3x + 7)^\circ \) are same - side interior angles? Wait, no, looking at the diagram, actually, if we consider the transversal, the angles \( (x + 5)^\circ \) and \( (3x + 7)^\circ \) are supplementary? Wait, no, maybe they are same - side interior angles? Wait, no, let's re - examine. Wait, actually, when two parallel lines are cut by a transversal, same - side interior angles are supplementary. Wait, but in the diagram, the angles \( (x + 5)^\circ \) and \( (3x + 7)^\circ \): Wait, no, maybe they are alternate interior angles? Wait, no, the positions: the upper angle is \( (x + 5)^\circ \) on line \( m \), the lower angle is \( (3x + 7)^\circ \) on line \( n \), and the transversal is cutting them. Wait, actually, if \( m\parallel n \), then the consecutive interior angles (same - side interior angles) are supplementary? Wait, no, maybe the angles are same - side interior angles? Wait, no, let's think again. Wait, the sum of same - side interior angles is \( 180^\circ \)? Wait, no, same - side interior angles are supplementary. Wait, but maybe the angles here are same - side interior angles? Wait, no, looking at the diagram, the two angles \( (x + 5)^\circ \) and \( (3x + 7)^\circ \): Wait, maybe they are supplementary? Wait, no, maybe I made a mistake. Wait, actually, when two parallel lines are cut by a transversal, the same - side interior angles are supplementary. Wait, but in the diagram, the angles \( (x + 5)^\circ \) and \( (3x + 7)^\circ \): Wait, no, maybe they are alternate exterior angles? No, alternate exterior angles are equal. Wait, maybe the angles are same - side interior angles, so \( (x + 5)+(3x + 7)=180 \)? Wait, no, that would be if they are same - side interior angles. Wait, but let's check the diagram again. Wait, the two angles are on the same side of the transversal, between the two parallel lines. So they are same - side interior angles, so their sum is \( 180^\circ \). Wait, no, wait, maybe I got it wrong. Wait, no, let's do the math. Wait, if \( m\parallel n \), and the transversal cuts them, then same - side interior angles are supplementary. So:

\( (x + 5)+(3x + 7)=180 \)

Step2: Solve the equation

Combine like terms:

\( x+3x + 5 + 7=180 \)

\( 4x+12 = 180 \)

Subtract 12 from both sides:

\( 4x=180 - 12 \)

\( 4x = 168 \)

Divide both sides by 4:

\( x=\frac{168}{4}=42 \)? Wait, no, that can't be. Wait, maybe the angles are alternate interior angles? Wait, alternate interior angles are equal. So \( x + 5=3x+7 \)? Wait, that would give \( 5 - 7 = 3x - x \), \( - 2 = 2x \), \( x=-1 \), which doesn't make sense. Wait, maybe I misidentified the angle relationship. Wait, maybe the angles are same - side exterior angles? No, same - side exterior angles are also supplementary. Wait, no, let's look at the diagram again. The upper angle is \( (x + 5)^\circ \), the lower angle is \( (3x + 7)^\circ \). Wait, maybe the angles are supplementary because they are same - side interior angles. Wait, but when I solved \( (x + 5)+(3x + 7)=180 \), I got \( 4x+12 = 180 \), \( 4x = 168 \), \( x = 42 \). But let's check the angle measures. \( x + 5=47 \), \( 3x + 7=133 \), and \( 47+133 = 180 \), which works. Wait, maybe that's correct. Wait, but maybe I made a mistake in the angle relationship. Wait, another way: if the lines are parallel, and the transversal cuts them, then consecutive interior angles (same - side interior angles) are supplementary. So that's the correct relationship here. So:

Step1: Set up the equation

Since \( m\parallel n \), the same - side interior angles are supplementary. So:

\( (x + 5)+(3x + 7)=180 \)

Step2: Simplify the left - hand side

Combine like terms:

\( x+3x+5 + 7=180 \)

\( 4x + 12=180 \)

Step3: Solve for x

Subtract 12 from both sides:

\( 4x=180 - 12 \)

\( 4x = 168 \)

Divide both sides by 4:

\( x=\frac{168}{4}=42 \)

Wait, but let's check again. Wait, maybe the angles are alternate interior angles? But if they were alternate interior angles, \( x + 5=3x + 7 \), which gives \( x=-1 \), which is impossible because angle measures can't be negative. So the correct relationship is that they are same - side interior angles, so their sum is \( 180^\circ \). So the value of \( x \) is 42.

Answer:

\( x = 42 \)