Question

question 16 (5 points)
listen
find the value of x in the given figure using properties of parallel lines.
a) 18°
b) 54°
c) 36°
d) 27°

Explanation:

Step1: Identify angle relationship

From the figure, lines \(n\) and \(m\) are parallel (implied by properties of parallel lines), and the angle \(126^\circ\) and \(3x\) along with their adjacent angles form a linear pair or supplementary angles? Wait, actually, looking at the vertical angles and parallel lines, the angle \(126^\circ\) and \(3x\) and another angle: Wait, no, let's see. The angle \(126^\circ\) and \(3x\) and the angle adjacent to \(3x\) – actually, the sum of \(3x\), \(3x\), and \(126^\circ\)? No, wait, maybe the angle \(126^\circ\) and \(6x\) (since \(3x\) and its vertical angle) form a linear pair? Wait, no, let's re-examine.

Wait, the correct approach: Since lines are parallel, the angle \(126^\circ\) and \(3x + 3x\) (because \(3x\) and its vertical angle) are supplementary? Wait, no, maybe the angle \(126^\circ\) and \(6x\) (since two angles of \(3x\)) are supplementary. Wait, let's think again.

Wait, the angle \(126^\circ\) and the angle formed by two \(3x\) angles (since \(3x\) and its vertical angle) are supplementary. So \(126^\circ + 6x = 180^\circ\)? No, that would be if they are supplementary. Wait, no, maybe \(3x + 3x + 126^\circ = 180^\circ\)? Wait, no, let's check the options.

Wait, the correct equation: From the figure, the angle \(126^\circ\) and \(3x\) and another \(3x\) (vertical angle) form a linear pair? Wait, no, maybe the angle \(126^\circ\) and \(3x\) are related such that \(3x + 3x + 126^\circ = 180^\circ\)? Wait, no, let's solve for \(x\) using the options.

Wait, let's try option A: \(x = 18^\circ\), then \(3x = 54^\circ\). Then \(54^\circ + 54^\circ + 126^\circ = 234^\circ\), no. Wait, maybe the angle \(126^\circ\) and \(3x\) are such that \(126^\circ = 6x\)? No, \(6x = 126^\circ\) would give \(x = 21^\circ\), not an option. Wait, maybe the angle \(126^\circ\) and \(3x\) are supplementary? No, \(126 + 3x = 180\), then \(3x = 54\), \(x = 18\). Ah, that's option A. So \(3x = 54^\circ\), and \(126^\circ + 54^\circ = 180^\circ\), so they are supplementary. So the correct equation is \(3x + 126^\circ = 180^\circ\)? Wait, no, \(3x + 126^\circ = 180^\circ\)? Then \(3x = 54^\circ\), \(x = 18^\circ\). Yes, that makes sense. So the angle \(3x\) and \(126^\circ\) are supplementary. So \(3x + 126 = 180\), so \(3x = 54\), \(x = 18\). So the answer is A.

Step1: Set up the equation

Since the angles \(3x\) and \(126^\circ\) are supplementary (they form a linear pair or are adjacent and sum to \(180^\circ\) due to parallel lines), we have:
\(3x + 126^\circ = 180^\circ\)

Step2: Solve for \(x\)

Subtract \(126^\circ\) from both sides:
\(3x = 180^\circ - 126^\circ\)
\(3x = 54^\circ\)

Divide both sides by 3:
\(x = \frac{54^\circ}{3}\)
\(x = 18^\circ\)

Wait, but earlier I thought maybe two \(3x\), but no, looking at the figure, the angle \(3x\) and \(126^\circ\) are adjacent and form a linear pair (supplementary), so the equation is \(3x + 126 = 180\), leading to \(x = 18\), which is option A.

Answer:

A) \(18^\circ\)