Question

an engineer is designing the parking lot shown in the diagram below for a local grocery store. the parking spaces are marked with lines where $overleftrightarrow{xd}paralleloverleftrightarrow{tz}paralleloverleftrightarrow{yf}paralleloverleftrightarrow{lp}$ and $overleftrightarrow{wc}$ is a transversal. in order for cars to fit in the spaces, the $mangle xrh$ must be a certain angle measure. if $mangle xrh = 7x + 5^{circ}$ and $mangle lnc = 10x - 16^{circ}$ what is $mangle xrh$?

Explanation:

Step1: Identify angle - relationship

Since $\overleftrightarrow{XD}\parallel\overleftrightarrow{TZ}\parallel\overleftrightarrow{YF}\parallel\overleftrightarrow{LP}$ and $\overleftrightarrow{WC}$ is a transversal, $\angle XRH$ and $\angle LNC$ are corresponding angles. Corresponding angles are congruent when lines are parallel, so $m\angle XRH=m\angle LNC$.

Step2: Set up the equation

Set $7x + 5=10x-16$.

Step3: Solve for $x$

Subtract $7x$ from both sides: $5 = 10x-7x - 16$, which simplifies to $5=3x - 16$. Then add 16 to both sides: $5 + 16=3x$, so $21 = 3x$. Divide both sides by 3: $x = 7$.

Step4: Find $m\angle XRH$

Substitute $x = 7$ into the expression for $m\angle XRH$. $m\angle XRH=7x + 5=7\times7+5=49 + 5=54^{\circ}$.

Answer:

$54^{\circ}$