Question

a diagram is shown where $overleftrightarrow{rg}paralleloverleftrightarrow{hm}$ with $overleftrightarrow{wz}$ and $overleftrightarrow{nd}$ as transversals. $mangle nkg=(9y + 9)^{circ}$, $mangle rbw = 56^{circ}$, $mangle dtz=z^{circ}$, and $angle htd=(8y + 16)^{circ}$. complete the statements. the value of $z$ is. the measure of $angle nkb$ is. the measure of $angle htd$ is.

Explanation:

Step1: Identify corresponding - angles

Since $\overrightarrow{RG}\parallel\overrightarrow{HM}$ and $\overrightarrow{WZ}$ is a transversal, $\angle RBW$ and $\angle HTD$ are corresponding angles. So, $m\angle HTD=m\angle RBW$.
$8y + 16=56$

Step2: Solve for $y$

Subtract 16 from both sides of the equation: $8y=56 - 16=40$. Then divide both sides by 8: $y = 5$.

Step3: Find $m\angle NKG$

Substitute $y = 5$ into the expression for $m\angle NKG$: $m\angle NKG=(9y + 9)^{\circ}=(9\times5+9)^{\circ}=(45 + 9)^{\circ}=54^{\circ}$.

Step4: Find $z$

$\angle NKG$ and $\angle DTZ$ are vertical - angles. Since vertical angles are equal, $z=m\angle NKG = 54$.

Step5: Find $m\angle NKB$

$\angle NKB$ and $\angle NKG$ are supplementary (linear - pair). So, $m\angle NKB=180 - m\angle NKG=180 - 54=126^{\circ}$.

Step6: Find $m\angle HTD$

We already found that $m\angle HTD = 56^{\circ}$ from the corresponding - angles property.

Answer:

The value of $z$ is $54$.
The measure of $\angle NKB$ is $126$.
The measure of $\angle HTD$ is $56$.