Question

in exercises 3–8, find the value of x that makes m ∥ n. explain your reasoning. (see example 1.) 3. image of two parallel lines m and n cut by a transversal, with 120° on line m and 3x° on line n 4. image of two parallel lines m and n cut by a transversal, with 135° on line m and (2x + 15)° on line n 5. image of two parallel lines m and n cut by a transversal, with 150° and (3x - 15)° 6. image of two parallel lines m and n cut by a transversal, with (180 - x)° on line m and x° on line n 7. image of two parallel lines m and n cut by a transversal, with x° and 2x° 8. image of two parallel lines m and n cut by a transversal, with 3x° and (2x + 20)°

Explanation:

Problem 3:

Step1: Identify Angle Relationship

For lines \( m \parallel n \), the \( 120^\circ \) angle and \( 3x^\circ \) angle are alternate exterior angles (or corresponding, depending on the transversal). Alternate exterior angles are equal when lines are parallel. So \( 3x = 120 \).

Step2: Solve for \( x \)

Divide both sides by 3: \( x = \frac{120}{3} = 40 \).

Step1: Identify Angle Relationship

For \( m \parallel n \), the \( 135^\circ \) angle and \( (2x + 15)^\circ \) angle are same - side interior angles? No, wait, they are supplementary (since they form a linear pair with parallel lines, consecutive interior angles? Wait, actually, the \( 135^\circ \) and \( (2x + 15)^\circ \) are same - side interior angles? No, let's see: when two parallel lines are cut by a transversal, consecutive interior angles are supplementary. Wait, the \( 135^\circ \) and \( (2x + 15)^\circ \) should be supplementary? Wait, no, actually, the angle adjacent to \( 135^\circ \) (linear pair) is \( 180 - 135=45^\circ \), but no, wait, the \( (2x + 15)^\circ \) and \( 135^\circ \) are same - side interior angles? Wait, no, let's think again. If \( m \parallel n \), then the angle \( (2x + 15)^\circ \) and the angle supplementary to \( 135^\circ \) (i.e., \( 45^\circ \))? No, wait, the correct relationship: the \( 135^\circ \) and \( (2x + 15)^\circ \) are same - side interior angles? No, actually, they are supplementary. Wait, \( 2x+15 + 135=180 \)? No, that would be if they are same - side interior angles. Wait, no, let's look at the diagram. The \( 135^\circ \) angle and \( (2x + 15)^\circ \) angle: when \( m \parallel n \), they are same - side interior angles, so they should be supplementary. So \( 2x + 15+135 = 180 \)? Wait, no, \( 2x + 15=180 - 135 \)? Wait, no, I made a mistake. The \( 135^\circ \) angle and \( (2x + 15)^\circ \) angle are actually alternate interior angles? No, wait, the correct approach: if \( m \parallel n \), then the angle \( (2x + 15)^\circ \) and the angle that is vertical to the supplementary angle of \( 135^\circ \)? No, let's start over. The sum of same - side interior angles is \( 180^\circ \). So \( (2x + 15)+135 = 180 \)? Wait, \( 2x+15 = 180 - 135=45 \), then \( 2x=30 \), \( x = 15 \)? Wait, no, that can't be. Wait, maybe they are alternate interior angles. Wait, the \( 135^\circ \) angle and \( (2x + 15)^\circ \) angle: if they are alternate interior angles, then \( 2x + 15=135 \)? No, that would make \( 2x = 120 \), \( x = 60 \), which is wrong. Wait, I think I messed up the angle relationship. Let's look at the diagram again. The line \( m \) and \( n \) are parallel, cut by a transversal. The \( 135^\circ \) angle and \( (2x + 15)^\circ \) angle: the \( 135^\circ \) angle and \( (2x + 15)^\circ \) angle are same - side interior angles, so they should be supplementary. So \( (2x + 15)+135 = 180 \). Then \( 2x+150 = 180 \), \( 2x = 30 \), \( x = 15 \). Wait, but let's check: \( 2x+15=2*15 + 15 = 45 \), and \( 135 + 45=180 \), which works for same - side interior angles. So that's correct.

Step1: Determine Angle Relationship

Since \( m\parallel n \), the angles \( (2x + 15)^\circ \) and \( 135^\circ \) are same - side interior angles, so they are supplementary. Thus, \( (2x + 15)+135=180 \).

Step2: Solve the Equation

Simplify the left - hand side: \( 2x+150 = 180 \). Subtract 150 from both sides: \( 2x=180 - 150 = 30 \). Divide both sides by 2: \( x = 15 \).

Step1: Identify Angle Relationship

For \( m\parallel n \), the \( 150^\circ \) angle and \( (3x - 15)^\circ \) angle are same - side interior angles? Wait, no, the \( 150^\circ \) angle and \( (3x - 15)^\circ \) angle: since \( m\parallel n \), the \( (3x - 15)^\circ \) angle and the supplementary angle of \( 150^\circ \) (i.e., \( 30^\circ \))? No, wait, the \( 150^\circ \) angle and \( (3x - 15)^\circ \) angle are same - side interior angles, so they should be supplementary. Wait, no, the \( 150^\circ \) angle and \( (3x - 15)^\circ \) angle: let's look at the diagram. The \( 150^\circ \) angle and \( (3x - 15)^\circ \) angle are same - side interior angles, so \( (3x - 15)+150 = 180 \)? No, that would be if they are same - side interior angles. Wait, no, the correct relationship: the \( 150^\circ \) angle and \( (3x - 15)^\circ \) angle are actually supplementary? Wait, \( 3x-15 + 150=180 \)? \( 3x + 135=180 \), \( 3x = 45 \), \( x = 15 \)? No, that can't be. Wait, maybe the \( (3x - 15)^\circ \) angle and the angle vertical to the angle supplementary to \( 150^\circ \). Wait, the angle supplementary to \( 150^\circ \) is \( 30^\circ \), and if \( m\parallel n \), then \( 3x - 15=30 \)? Then \( 3x=45 \), \( x = 15 \). Wait, but let's check: \( 3x - 15=3*15 - 15 = 30 \), and \( 150+30 = 180 \), which is correct for same - side interior angles. So that's the relationship.

Step1: Angle Relationship

Since \( m\parallel n \), \( (3x - 15)^\circ \) and \( 150^\circ \) are same - side interior angles, so \( (3x - 15)+150 = 180 \).

Step2: Solve for \( x \)

Simplify: \( 3x+135 = 180 \). Subtract 135: \( 3x=180 - 135 = 45 \). Divide by 3: \( x = 15 \).

Answer:

\( x = 40 \)

Problem 4: