Question

if (mangle e + mangle f=120^{circ}), which expression is equivalent to (mangle g)?
options:
a. ((7x)^{circ})
b. ((7x + 4)^{circ})
c. ((8x)^{circ})
d. ((8x + 4)^{circ})

Explanation:

Step1: Find sum of ∠E and ∠F

$m\angle E+m\angle F=(7x + 2)+(8x - 2)=15x$

Step2: Set up equation using given sum

Since $m\angle E+m\angle F = 120^{\circ}$, we have $15x=120$. Solving for $x$, we get $x = 8$.

Step3: Use angle - sum property of a triangle

The sum of angles in a triangle is $180^{\circ}$. Let $m\angle G=y$. Then $m\angle E+m\angle F + y=180^{\circ}$. Substituting $m\angle E+m\angle F = 120^{\circ}$, we get $y=180-(m\angle E+m\angle F)$. Also, since $m\angle E+m\angle F = 15x$ and $x = 8$, we know $m\angle E+m\angle F=120^{\circ}$. Now, we can also express $m\angle G$ in terms of $x$. The sum of angles in $\triangle EFG$ is $180^{\circ}$, so $m\angle G=180-(7x + 2)-(8x - 2)=180 - 7x-2 - 8x + 2=180-15x$. Substituting $x = 8$, we can check our work. Another way is to note that $m\angle G=180-(m\angle E+m\angle F)$. Since $m\angle E+m\angle F = 15x$ and we want to find $m\angle G$ in terms of $x$. We know that $m\angle G=180-(7x + 2+8x - 2)=180 - 15x$. If we consider the relationship between the given angles and the sum of angles in a triangle, we can rewrite $m\angle G$ as follows:
\[

$$\begin{align*} m\angle G&=180-(7x + 2+8x - 2)\\ &=180-(15x)\\ \end{align*}$$

\]
We know $m\angle E+m\angle F = 15x=120^{\circ}$. Now, $m\angle G=180 - 15x$. If we want to express it in a non - subtracted form, we know that $m\angle E+m\angle F = 15x$ and the sum of angles in a triangle is $180^{\circ}$. So $m\angle G=180-(m\angle E+m\angle F)$. Since $m\angle E+m\angle F=(7x + 2)+(8x - 2)=15x$, we can also say that $m\angle G = 180-15x$. If we rewrite it in terms of the options, we know that $m\angle G=180-(7x + 2+8x - 2)=180-15x$. Now, $m\angle G = 60^{\circ}$. Substituting $x = 8$ into the expressions for the options:

• Option A: $(7x)^{\circ}=7\times8 = 56^{\circ}$
• Option B: $(7x + 4)^{\circ}=7\times8+4=60^{\circ}$
• Option C: $(8x)^{\circ}=8\times8 = 64^{\circ}$
• Option D: $(8x + 4)^{\circ}=8\times8+4=68^{\circ}$

Answer:

B. $(7x + 4)^{\circ}$