Question

1. what is the value of x which will allow you to prove that \\(\overline{ab} \parallel \overline{cd}\\) when \\(m\angle f = 144^\circ\\) and \\(m\angle e = (3x - 30)^\circ\\)?\

a 22\
b 36\
c 58\
d 66

Explanation:

Step1: Identify the relationship between angles

To prove \( \overline{AB} \parallel \overline{CD} \), \( \angle F \) and \( \angle E \) should be supplementary (since they are same - side interior angles or corresponding supplementary angles for parallel lines). So, \( m\angle F + m\angle E=180^{\circ} \).

Step2: Substitute the given angle measures

We know that \( m\angle F = 144^{\circ} \) and \( m\angle E=(3x - 30)^{\circ} \). Substituting these into the supplementary angle equation, we get:
\( 144+(3x - 30)=180 \)

Step3: Simplify the equation

First, simplify the left - hand side of the equation:
\( 144+3x - 30 = 180 \)
\( 114+3x=180 \)

Step4: Solve for x

Subtract 114 from both sides of the equation:
\( 3x=180 - 114 \)
\( 3x = 66 \)
Then divide both sides by 3:
\( x=\frac{66}{3}=22 \)? Wait, no, wait. Wait, maybe I made a mistake in the angle relationship. Wait, looking at the diagram, \( \angle F \) and \( \angle E \) are actually corresponding angles or maybe alternate exterior? Wait, no, let's re - examine. Wait, if \( AB\parallel CD \), then \( \angle F \) and \( \angle E \) should be equal? No, wait, no. Wait, maybe \( \angle F \) and \( \angle E \) are same - side interior angles? Wait, no, let's check the diagram again. The transversal cuts \( AB \) at \( F \) and \( CD \) at \( E \). So \( \angle F \) (at \( AB \)) and \( \angle E \) (at \( CD \)): if \( AB\parallel CD \), then \( \angle F \) and \( \angle E \) are equal? Wait, no, maybe I got the relationship wrong. Wait, actually, if we consider the vertical angles or corresponding angles. Wait, no, let's start over.

Wait, the correct relationship: If \( AB\parallel CD \), then \( \angle F \) and \( \angle E \) are supplementary? Wait, no, let's calculate again.

Wait, the equation \( 144+(3x - 30)=180 \)
\( 3x+114 = 180 \)
\( 3x=180 - 114=66 \)
\( x = 22 \)? But 22 is option A. Wait, but let's check: if \( x = 22 \), then \( m\angle E=3\times22 - 30=66 - 30 = 36^{\circ} \), and \( 144 + 36=180 \), which means they are supplementary. So that makes sense for same - side interior angles, so \( AB\parallel CD \) by the converse of the same - side interior angles theorem.

Answer:

A. 22