Question

1. multiple choice 1 point 16. diagram: two vertical parallel lines cut by a transversal, with a 102° angle and a ((2x + 12)^circ) angle find the measure of ( x )

a) ( x = 33^circ )

b) ( x = 39^circ )

c) ( x = 66^circ )

d) ( x = 45^circ )

Explanation:

Step1: Identify the relationship

The two angles are corresponding angles (since the lines are parallel), so they are equal? Wait, no, wait. Wait, the 102° and (2x + 12)°—wait, actually, if the lines are parallel, and the transversal cuts them, then the 102° and (2x + 12)°—wait, maybe they are alternate interior angles or corresponding? Wait, no, actually, 102° and (2x + 12)°—wait, maybe they are same - side? No, wait, looking at the diagram, the two vertical lines are parallel, and the transversal crosses them. So the angle 102° and (2x + 12)°—wait, actually, 102° and (2x + 12)° should be equal? Wait, no, wait, 102° and (2x + 12)°—wait, maybe 102° and (2x + 12)° are equal? Wait, no, let's think again. Wait, if the two lines are parallel, then the corresponding angles are equal. Wait, the angle 102° and (2x + 12)°—wait, maybe 102° and (2x + 12)° are equal? Wait, no, let's set up the equation. Wait, actually, 102° and (2x + 12)°—wait, maybe they are equal? Wait, no, wait, 102 = 2x + 12? Wait, no, that would give 2x = 90, x = 45, but that's not right. Wait, no, maybe they are supplementary? Wait, no, 102 + (2x + 12)=180? Wait, 102 + 2x + 12 = 180 → 2x + 114 = 180 → 2x = 66 → x = 33? Wait, no, that's not. Wait, no, let's look at the diagram again. The two vertical lines are parallel, and the transversal. The angle 102° and (2x + 12)°—wait, maybe (2x + 12)° is equal to 102°? No, that would be if they are corresponding. Wait, no, maybe I made a mistake. Wait, let's check the options. Let's try x = 33: 2*33 + 12 = 78, 102 + 78 = 180, so they are supplementary. Ah, right! Because the two angles are same - side interior angles? Wait, no, same - side interior angles are supplementary. Wait, the two vertical lines are parallel, so the consecutive interior angles are supplementary. So 102 + (2x + 12)=180.

Step2: Solve the equation

So, 102+(2x + 12)=180.
First, simplify the left - hand side: 2x+114 = 180.
Then, subtract 114 from both sides: 2x=180 - 114=66.
Then, divide both sides by 2: x = 33.

Answer:

a) \( x = 33^{\circ} \)