Question

solve for x to make a||b. a (3x + 3)° b (16x + 6)° x = ?

Explanation:

Step1: Identify the relationship

When two lines \( A \) and \( B \) are parallel, the consecutive interior angles (or in this case, the angle \( (3x + 3)^\circ \) and the supplementary angle to \( (16x + 6)^\circ \)) should be supplementary? Wait, actually, looking at the diagram, the angle \( (3x + 3)^\circ \) and the angle \( (16x + 6)^\circ \) are same - side exterior - interior? Wait, no, when \( A\parallel B \), the angle \( (3x + 3)^\circ \) and the angle \( (16x + 6)^\circ \) are same - side interior? Wait, no, actually, the angle \( (3x + 3)^\circ \) and the angle \( (16x + 6)^\circ \) are same - side and should be supplementary? Wait, no, let's think again. The angle \( (3x + 3)^\circ \) and the angle \( (16x + 6)^\circ \) are same - side and when \( A\parallel B \), they are same - side interior angles? Wait, no, actually, the angle \( (3x + 3)^\circ \) and the angle \( (16x + 6)^\circ \) are same - side and should be supplementary? Wait, no, let's correct. When two parallel lines are cut by a transversal, same - side interior angles are supplementary. But in this case, the angle \( (3x + 3)^\circ \) and the angle \( (16x + 6)^\circ \) are same - side and actually, \( (3x + 3)+(16x + 6)=180 \)? Wait, no, wait. Wait, the angle \( (16x + 6)^\circ \) and its adjacent angle (on line B) are supplementary. But when \( A\parallel B \), the angle \( (3x + 3)^\circ \) is equal to the adjacent angle of \( (16x + 6)^\circ \)? No, wait, let's look at the diagram again. The angle \( (3x + 3)^\circ \) and the angle \( (16x + 6)^\circ \): when \( A\parallel B \), the angle \( (3x + 3)^\circ \) and the angle \( (16x + 6)^\circ \) are same - side and should be supplementary? Wait, no, actually, the correct relationship is that \( (3x + 3)+(16x + 6)=180 \)? Wait, no, let's calculate. Wait, if \( A\parallel B \), then the angle \( (3x + 3)^\circ \) and the angle \( (16x + 6)^\circ \) are same - side interior angles? Wait, no, maybe I made a mistake. Wait, the angle \( (3x + 3)^\circ \) and the angle \( (16x + 6)^\circ \): let's check the sum. \( 3x+3 + 16x + 6=19x + 9 \). If they are supplementary, \( 19x+9 = 180 \), then \( 19x=171 \), \( x = 9 \). Oh, that works. So the correct equation is \( (3x + 3)+(16x + 6)=180 \) because when two parallel lines are cut by a transversal, same - side interior angles are supplementary.

Step2: Solve the equation

We have the equation \( 3x+3 + 16x+6 = 180 \).
Combine like terms: \( (3x+16x)+(3 + 6)=180 \), which simplifies to \( 19x+9 = 180 \).
Subtract 9 from both sides: \( 19x=180 - 9=171 \).
Divide both sides by 19: \( x=\frac{171}{19}=9 \).

Answer:

9