Question

45° and 3x are angles formed by intersecting lines. notes:

Explanation:

Step1: Identify angle relationship

The two angles \(45^\circ\) and \(3x\) are adjacent and form a linear pair? No, wait, actually, when two lines intersect, adjacent angles on a straight line sum to \(180^\circ\)? Wait, no, looking at the diagram, the two angles \(45^\circ\) and \(3x\) are adjacent and form a linear pair? Wait, no, actually, if we consider the straight line, the sum of angles on a straight line is \(180^\circ\). Wait, but also, vertical angles? Wait, no, in the diagram, the two angles \(45^\circ\) and \(3x\) are adjacent and form a linear pair? Wait, no, let's re-examine. When two lines intersect, the adjacent angles are supplementary (sum to \(180^\circ\))? Wait, no, actually, in this case, the angle \(45^\circ\) and \(3x\) are adjacent and form a linear pair? Wait, no, maybe they are supplementary? Wait, no, wait, if we look at the intersection, the angle \(45^\circ\) and \(3x\) are adjacent and form a linear pair, so their sum should be \(180^\circ\)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, actually, when two lines intersect, the adjacent angles are supplementary. Wait, but in this case, the angle \(45^\circ\) and \(3x\) are adjacent, so \(45 + 3x = 180\)? Wait, no, that would give \(3x = 135\), \(x = 45\), but that seems off. Wait, no, maybe the two angles are vertical angles? No, vertical angles are equal. Wait, no, the diagram shows two intersecting lines, with one angle \(45^\circ\) and the adjacent angle \(3x\). Wait, maybe the sum of \(45^\circ\) and \(3x\) is \(180^\circ\) because they are supplementary (form a linear pair). Let's check:

Step2: Set up the equation

Since the two angles \(45^\circ\) and \(3x\) are supplementary (they form a linear pair, so their sum is \(180^\circ\)), we can write the equation:
\(45 + 3x = 180\)

Step3: Solve for \(x\)

Subtract \(45\) from both sides:
\(3x = 180 - 45\)
\(3x = 135\)

Divide both sides by \(3\):
\(x = \frac{135}{3}\)
\(x = 45\)

Wait, but that seems like \(3x = 135\), so \(x = 45\). But let's verify. If \(x = 45\), then \(3x = 135\), and \(45 + 135 = 180\), which is correct for supplementary angles.

Answer:

\(x = 45\)