Question

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

Explanation:

Step1: Identify vertical - angle relationship

Since $\angle RBW$ and $\angle HTD$ are vertical angles, $\angle RBW=\angle HTD$. So, $56 = 8y + 16$.

Step2: Solve for $y$

Subtract 16 from both sides: $56-16=8y$, which gives $40 = 8y$. Then divide both sides by 8: $y=\frac{40}{8}=5$.

Step3: Find $\angle NKG$

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

Step4: Identify corresponding - angle relationship

$\angle DTZ$ and $\angle NKG$ are corresponding angles. Since $RG\parallel HM$, $\angle DTZ=\angle NKG$. So, $x = 54$.

Step5: Find $\angle NKB$

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

Step6: Find $\angle HTD$

We already know from the vertical - angle relationship that $\angle HTD = 56^{\circ}$.

Answer:

The value of $x$ is $54$.
The measure of $\angle NKB$ is $126^{\circ}$.
The measure of $\angle HTD$ is $56^{\circ}$.