Question

m∠efq = 2x + 18, m∠efg = 17x + 8, and m∠qfg = 140°. find m∠efg. on this problem, you are asked to find the measure of an angle. this means you must plug your x into that ang

Explanation:

Step1: Identify angle relationship

From the diagram, \( \angle EFG = \angle EFQ + \angle QFG \) is incorrect. Actually, \( \angle EFG = \angle EFQ + \angle QFG \) is wrong. Wait, looking at the diagram, points \( E, F, G \) are colinear (since \( G \) and \( E \) are on a straight line through \( F \)), and \( QF \) is a ray from \( F \). So actually, \( \angle EFG \) is a straight angle? No, wait, \( \angle EFQ + \angle QFG=\angle EFG \)? Wait, no, if \( E - F - G \) is a straight line, then \( \angle EFG \) is a straight angle? Wait, no, the given \( m\angle QFG = 140^\circ \), and \( \angle EFQ \) and \( \angle QFG \) are adjacent angles forming \( \angle EFG \)? Wait, no, maybe \( \angle EFG=\angle EFQ + \angle QFG \)? Wait, no, let's re - examine. The correct relationship: since \( E, F, G \) are on a straight line, \( \angle EFG \) is a straight angle? No, the problem says \( m\angle QFG = 140^\circ \), \( m\angle EFQ=2x + 18 \), \( m\angle EFG = 17x+8 \). So the correct angle addition postulate is \( \angle EFG=\angle EFQ+\angle QFG \)? Wait, no, that would mean \( 17x + 8=(2x + 18)+140 \). Let's check:

Step2: Set up the equation

Using the angle addition postulate (if \( \angle EFQ \) and \( \angle QFG \) are adjacent angles with a common side \( FQ \) and their non - common sides \( FE \) and \( FG \) form \( \angle EFG \)), then \( m\angle EFG=m\angle EFQ + m\angle QFG \)

So we set up the equation:
\( 17x + 8=(2x + 18)+140 \)

Step3: Solve for \( x \)

First, simplify the right - hand side:
\( 2x+18 + 140=2x+158 \)

So the equation becomes \( 17x + 8=2x+158 \)

Subtract \( 2x \) from both sides:
\( 17x-2x + 8=2x-2x+158 \)
\( 15x + 8=158 \)

Subtract 8 from both sides:
\( 15x+8 - 8=158 - 8 \)
\( 15x=150 \)

Divide both sides by 15:
\( x=\frac{150}{15}=10 \)

Step4: Find \( m\angle EFG \)

We know that \( m\angle EFG = 17x+8 \). Substitute \( x = 10 \) into the expression:

\( m\angle EFG=17\times10 + 8=170 + 8=178^\circ \)

Wait, but let's check the angle relationship again. Wait, maybe I made a mistake in the angle addition. If \( E, F, G \) are colinear, then \( \angle EFG \) is a straight angle (\( 180^\circ \))? But \( 178^\circ \) is close to \( 180^\circ \), and \( \angle QFG = 140^\circ \), \( \angle EFQ=2x + 18=2\times10+18 = 38^\circ \), and \( 38^\circ+140^\circ = 178^\circ \), which is \( \angle EFG \). So that seems correct.

Answer:

\( 178^\circ \)