Question

given ( m parallel n ), find the value of ( x ).

Explanation:

Step1: Identify Angle Relationship

Since \( m \parallel n \), the angles \( (x - 3)^\circ \) and \( (4x + 3)^\circ \) are same - side interior angles? Wait, no, looking at the diagram, actually, the angle \( (x - 3)^\circ \) and the angle adjacent to \( (4x + 3)^\circ \) (vertical angles or corresponding? Wait, no, let's re - examine. Wait, when two parallel lines are cut by a transversal, same - side interior angles are supplementary? Wait, no, actually, the angle \( (x - 3)^\circ \) and \( (4x + 3)^\circ \) are alternate exterior angles? No, wait, maybe they are same - side interior angles? Wait, no, let's think again. If \( m\parallel n \), and the transversal cuts them, the angle \( (x - 3)^\circ \) and \( (4x + 3)^\circ \) are actually supplementary? Wait, no, maybe they are alternate interior angles? Wait, no, let's check the positions. Wait, the angle \( (x - 3)^\circ \) is on line \( m \), above the transversal, and \( (4x + 3)^\circ \) is on line \( n \), below the transversal, but actually, they are same - side interior angles? Wait, no, maybe I made a mistake. Wait, actually, when two parallel lines are cut by a transversal, same - side interior angles are supplementary. Wait, but let's see: the angle \( (x - 3)^\circ \) and the angle that is supplementary to \( (4x + 3)^\circ \)? No, wait, maybe the angles \( (x - 3)^\circ \) and \( (4x + 3)^\circ \) are same - side interior angles, so they should add up to \( 180^\circ \)? Wait, no, that can't be. Wait, maybe they are alternate exterior angles? No, alternate exterior angles are equal. Wait, maybe the angle \( (x - 3)^\circ \) and \( (4x + 3)^\circ \) are same - side interior angles, so \( (x - 3)+(4x + 3)=180 \)? Wait, no, that would simplify to \( 5x = 180 \), \( x = 36 \), but that seems off. Wait, maybe I got the angle relationship wrong. Wait, actually, the angle \( (x - 3)^\circ \) and \( (4x + 3)^\circ \) are same - side interior angles? Wait, no, let's look at the diagram again. The two angles: one is \( (x - 3)^\circ \) on line \( m \), above the transversal, and \( (4x + 3)^\circ \) on line \( n \), below the transversal, but actually, they are same - side interior angles, so they are supplementary. Wait, but let's check the calculation. Wait, \( (x - 3)+(4x + 3)=180 \), \( x - 3+4x + 3 = 180 \), \( 5x=180 \), \( x = 36 \). Wait, but maybe they are alternate interior angles? No, alternate interior angles are equal. Wait, maybe the angle \( (x - 3)^\circ \) and \( (4x + 3)^\circ \) are supplementary? Wait, let's re - evaluate. If \( m\parallel n \), and the transversal cuts them, same - side interior angles are supplementary. So the angle \( (x - 3)^\circ \) and \( (4x + 3)^\circ \) are same - side interior angles, so their sum is \( 180^\circ \).

Step2: Solve the Equation

We set up the equation:
\( (x - 3)+(4x + 3)=180 \)
Simplify the left - hand side:
\( x-3 + 4x+3=5x \)
So we have \( 5x = 180 \)
Divide both sides by 5:
\( x=\frac{180}{5}=36 \)

Wait, but that seems incorrect. Wait, maybe the angles are alternate exterior angles? If they are alternate exterior angles, then \( x - 3=4x + 3 \), which would give \( - 3x=6 \), \( x=-2 \), which is impossible because angle measures can't be negative. So that's not it. Wait, maybe the angles are same - side interior angles. Wait, but let's check again. Wait, maybe the angle \( (x - 3)^\circ \) and \( (4x + 3)^\circ \) are supplementary. So \( x - 3+4x + 3 = 180 \), \( 5x=180 \), \( x = 36 \). But let's verify. If \( x = 36 \), then \( x - 3=33^\circ \), \( 4x + 3=4\times36+3 = 147^\circ \), and \( 33 + 147=180 \), which works. So the equation is correct.

Answer:

\( x = 36 \)