Question

find the measure of each angle. (angle efg) and (angle lmn) are supplementary angles, (mangle efg=(3x + 17)^{circ}), and (mangle lmn=(\frac{1}{2}x - 5)^{circ}). (mangle efg=square^{circ}) (mangle lmn=square^{circ})

Explanation:

Step1: Use supplementary - angle property

Since supplementary angles add up to 180°, we set up the equation $(3x + 17)+(\frac{1}{2}x-5)=180$.

Step2: Combine like - terms

Combine the x - terms and the constant terms: $3x+\frac{1}{2}x+17 - 5=180$.
$3x+\frac{1}{2}x=\frac{6x + x}{2}=\frac{7x}{2}$, and $17-5 = 12$. So the equation becomes $\frac{7x}{2}+12=180$.

Step3: Isolate the x - term

Subtract 12 from both sides of the equation: $\frac{7x}{2}=180 - 12$.
$\frac{7x}{2}=168$.

Step4: Solve for x

Multiply both sides by $\frac{2}{7}$: $x=168\times\frac{2}{7}$.
$x = 48$.

Step5: Find the measure of $\angle EFG$

Substitute $x = 48$ into the expression for $m\angle EFG$: $m\angle EFG=(3x + 17)^{\circ}=(3\times48+17)^{\circ}$.
$m\angle EFG=(144 + 17)^{\circ}=161^{\circ}$.

Step6: Find the measure of $\angle LMN$

Substitute $x = 48$ into the expression for $m\angle LMN$: $m\angle LMN=(\frac{1}{2}x-5)^{\circ}=(\frac{1}{2}\times48-5)^{\circ}$.
$m\angle LMN=(24 - 5)^{\circ}=19^{\circ}$.

Answer:

$m\angle EFG = 161^{\circ}$
$m\angle LMN = 19^{\circ}$