Question

a diagram is shown where $overleftrightarrow{hn}paralleloverleftrightarrow{mx}$ with $overleftrightarrow{wk}$ as a transversal. $mangle hrg=(11x + 12.9)^{circ}$, and $mangle xgr=(23x - 36.3)^{circ}$. what is $mangle hrg$? if necessary, round to the nearest whole number. 29° 34° 58° 79°

Explanation:

Step1: Identify angle - relationship

Since $\overleftrightarrow{HN}\parallel\overleftrightarrow{MX}$ and $\overleftrightarrow{WK}$ is a transversal, $\angle HRG$ and $\angle XGR$ are same - side interior angles. Same - side interior angles are supplementary, so $m\angle HRG + m\angle XGR=180^{\circ}$.

Step2: Set up the equation

Substitute the given expressions for the angle measures: $(11x + 12.9)+(23x-36.3)=180$.

Step3: Combine like - terms

$11x+23x+12.9 - 36.3=180$, which simplifies to $34x-23.4 = 180$.

Step4: Solve for $x$

Add 23.4 to both sides of the equation: $34x=180 + 23.4=203.4$. Then $x=\frac{203.4}{34}\approx5.98$.

Step5: Find $m\angle HRG$

Substitute $x$ into the expression for $m\angle HRG$: $m\angle HRG=11x + 12.9$. So $m\angle HRG=11\times5.98+12.9=65.78+12.9 = 78.68\approx79^{\circ}$.

Answer:

$79^{\circ}$